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All Textbook Solutions for Calculus Volume 3

In the following exercises, consider a lamina occupying the region R and having the density function p given in the preceding group of exercises. Use a computer algebra system (CAS) to answer the following questions. a. Find the moments M and M. about the x-axis and v-ax is, respectively. b. Calculate and plot the center of mass of the lamina. C. [T] Use a CAS to locate the center of mass on the graph of R. 316. [T] R is the unit disk; (x,y)=3x4+6x2y2+3y4 In the following exercises, consider a lamina occupying the region R and having the density function p given in the preceding group of exercises. Use a computer algebra system (CAS) to answer the following questions. a. Find the moments M and M. about the x-axis and v-ax is, respectively. b. Calculate and plot the center of mass of the lamina. C. [T] Use a CAS to locate the center of mass on the graph of R. 317. [T] R is the legion enclosed by the ellipse x2+4y=1;(x,y)=1 In the following exercises, consider a lamina occupying the region R and having the density function p given in the preceding group of exercises. Use a computer algebra system (CAS) to answer the following questions. a. Find the moments M and M. about the x-axis and y-ax is, respectively. b. Calculate and plot the center of mass of the lamina. C. [T] Use a CAS to locate the center of mass on the graph of R. 318. [T]=R={(x,y)9x2+y21,x0,y0};(x,y)=9x2+y2 .In the following exercises, consider a lamina occupying the region R and having the density function p given in the preceding group of exercises. Use a computer algebra system (CAS) to answer the following questions. a. Find the moments M and M. about the x-axis and y-ax is, respectively. b. Calculate and plot the center of mass of the lamina. C. [T] Use a CAS to locate the center of mass on the graph of R. 319. [T] R is the region bounded by y=x,y=x,y=x+2, and y=x+2;(x,y)=1 In the following exercises, consider a lamina occupying the region R and having the density function p given in the preceding group of exercises. Use a computer algebra system (CAS) to answer the following questions. a. Find the moments M and M. about the x-axis and y-ax is, respectively. b. Calculate and plot the center of mass of the lamina. C. [T] Use a CAS to locate the center of mass on the graph of R. 320. [T] R is the region bounded by y=1x,y=2x,y=1 and y=2;(x,y)=4(x+y).In the following exercises, consider a lamina occupying the region R and having the density function given in the first two groups of Exercises. a. Find the moments of inertia lx,lyand lo about the x-axis. y-axis, and origin, respectively. b. Find the radii of gyration with respect to the x-axis, y-axis, and origin, respectively. 32 1. R is the triangular region with vertices (0. 0). (0. 3). and (6. 0): p(x. v) = xv.In the following exercises, consider a lamina occupying the region R and having the density function given in the first two groups of Exercises. a. Find the moments of inertia lx,lyand lo about the x-axis. y-axis, and origin, respectively. b. Find the radii of gyration with respect to the x-axis, y-axis, and origin, respectively. 322. R is the triangular legion with vertices (0, 0), (1, 1), and (0, 5): (x. y) = x + y.In the following exercises, consider a lamina occupying the region R and having the density function given in the first two groups of Exercises. a. Find the moments of inertia lx,lyand lo about the x-axis. y-axis, and origin, respectively. b. Find the radii of gyration with respect to the x-axis, y-axis, and origin, respectively.In the following exercises, consider a lamina occupying the region R and having the density function given in the first two groups of Exercises. a. Find the moments of inertia lx,lyand lo about the x-axis. y-axis, and origin, respectively. b. Find the radii of gyration with respect to the x-axis, y-axis, and origin, respectively. 324. R is the rectangular region with vertices (0. 1), (0, 3), (3, 3), and (3, 1): (x, y) = x2y.In the following exercises, consider a lamina occupying the region R and having the density function given in the first two groups of Exercises. a. Find the moments of inertia lx,lyand lo about the x-axis. y-axis, and origin, respectively. b. Find the radii of gyration with respect to the x-axis, y-axis, and origin, respectively. 325. R is the trapezoidal region determined by the lines y=14x+52,y=0,y=2 . and x=0;(x,y)=2x+y In the following exercises, consider a lamina occupying the region R and having the density function given in the first two groups of Exercises. a. Find the moments of inertia lx,lyand lo about the x-axis. y-axis, and origin, respectively. b. Find the radii of gyration with respect to the x-axis, y-axis, and origin, respectively. 326. R is the trapezoidal region determined by the lines y=0,y=1,y=x and y=x+3;(x,y)=2x+y .In the following exercises, consider a lamina occupying the region R and having the density function given in the first two groups of Exercises. a. Find the moments of inertia lx,lyand lo about the x-axis. y-axis, and origin, respectively. b. Find the radii of gyration with respect to the x-axis, y-axis, and origin, respectively. 327. R is the disk of radius 2 centered at (1, 2): (x,y)=x2+y22x+5.In the following exercises, consider a lamina occupying the region R and having the density function given in the first two groups of Exercises. a. Find the moments of inertia lx,lyand lo about the x-axis. y-axis, and origin, respectively. b. Find the radii of gyration with respect to the x-axis, y-axis, and origin, respectively. 328. R is the unit disk: (x,y)=3x4+6x2y2+3y4 In the following exercises, consider a lamina occupying the region R and having the density function given in the first two groups of Exercises. a. Find the moments of inertia lx,lyand lo about the x-axis. y-axis, and origin, respectively. b. Find the radii of gyration with respect to the x-axis, y-axis, and origin, respectively. 329. R is the region enclosed by the ellipse x2+4y2=1;(x,y)=1.In the following exercises, consider a lamina occupying the region R and having the density function given in the first two groups of Exercises. a. Find the moments of inertia lx,lyand lo about the x-axis. y-axis, and origin, respectively. b. Find the radii of gyration with respect to the x-axis, y-axis, and origin, respectively. 330. R={(x,y)9x2+y21,x0,y0};(x,y)=9x2+y2 In the following exercises, consider a lamina occupying the region R and having the density function given in the first two groups of Exercises. a. Find the moments of inertia lx,lyand lo about the x-axis. y-axis, and origin, respectively. b. Find the radii of gyration with respect to the x-axis, y-axis, and origin, respectively. 331. R is the region bounded by y=x,y=x,y=x+2 . and y=x+2 p(x. y) = 1.In the following exercises, consider a lamina occupying the region R and having the density function given in the first two groups of Exercises. a. Find the moments of inertia lx,lyand lo about the x-axis. y-axis, and origin, respectively. b. Find the radii of gyration with respect to the x-axis, y-axis, and origin, respectively. 332. R is the region bounded y=1x,y=2x,y=1 , and y=2;(x,y)=4(x+y).Let Q be the solid unit cube. Find the mass of the solid if its density is equal to the square of the distance of an arbitrary point of Q to the xy-plane.Let Q be the solid unit hemisphere. Find the mass of the solid if its density is proportional to the distance of an arbitrary point of Q to the origin.The solid Q of constant density I is situated inside the sphere x2+y2+z2=16 and outside the sphere x2+y2+z2=1 . Show that the center of mass of the solid is not located within the solid.Find the mass of the solid Q={(x,y,z)1x2+z225,y1x2z2}whose density is (x,y,z). where k >0.Consider the solid Q={(x,y,z)0x1,0y2,0z3} with the density function (x,y,z)=x+y+1 . a. Find the mass of Q. b. Find the moments Mxy.. and Mxzabout the Myz-plane. xy-plane, and yz-plane, respectively. c. Find the center of mass of Q.[T] The solid Q has the mass given by the triple N integral 110401 r 2drddz.. Use a CAS to answer the following questions. a. Show that the center of mass of Q is located in the xy- plane. b. Graph Q and locate its center of mass.The solid Q is bounded by the planes x+4y+z=8,x=0,y=0 . And z=0. Its density at any point is equal to the distance to the xz-plane. Find the moments of inertia I of the solid about the xz-plane.The solid Q is bounded by the planes x+y+z=3 . and z = 0. Its density is p(x. y. z) = x + ay. where a > 0. Show that the center of mass of the solid is located in the plane z=35 for any value of a.Let Q be the solid situated outside the sphere x2+y2+z2=zand inside the upper hemisphere x2+y2+z2=R2. where R>1. If the density of the solid is (x,y,z)=1x2+y2+z2find R such that the 72 .The mass of a solid is given by 0f20f4x2 x 2+ y 2f16x2y2(x2+y2+x2)dzdydx, where n is an integer. Determine n such the mass of the solid is (22) .Let Q be the solid bounded above the cone x2+y2=z2 and below the sphere x2+y2+z24z=0 . Its density is a constant k > 0. Find k such that the center of mass of the solid is situated 7 units from the origin.The solid Q={(x,y,z)0x2+y216,x0,y0,0zx} has the density p(x. y. z) = k. Show that the moment about the xy-plane is half of the moment Myzabout the yz-p1ane.The solid Q is bounded by the cylinder + = a2. the paraboloid b2z=x2+y2, and the xy-plane. where 0 (x,y,z)=x2+y2 Let Q be a solid of constant density k. where k > 0. that is located in the first octant, inside the circular cone x2+ = 9(z — 1 )2 and above the plane z = 0. Show that the moment Mxy about the xy-plane is the same as the moment Myzabout the xz-plane.The solid Q has the mass given by the triple integral /2/6/6022(r4+r)dzddr . a. Find the density of the solid in rectangular coordinates. b. Find the moment M. about the xy-plane.The solid Q has the moment of inertia Ixabout yz-plane given by the triple integral 02 4 y 1 1 2 ( x 2+ y 2 ) 4 y 1 ( y 2+ z 2)( x 2+ y 2)dzdxdya. Find the density of Q. b. Find the moment of inertia Izabout the xy-plane.The solid Q has the mass given by the triple integral 0/402sec01( r3 cos+2r)dzdrd. a. Find the density of the solid in rectangular coordinates. b. Find the moment Mxzabout the xz-plane.A solid Q has a volume given by DabdAdz. where D is the projection of the solid onto the xy-plane and a < b are teal numbers, and its density does not depend on the variable z. Show that its center of mass lies in lie plane z=a+b2 .Consider the solid enclosed by the cylinder x2z2=a2 and the planes y=band y=c. where a0 and b < c are real numbers. The density of Q is given by (x,y,z)=f(y). where f is a differential function whose derivative is continuous on (b, c). Show that if f(b) = f(c). then the moment of inertia about the xz-plane of Q is null.[T] The average density of a solid Q is defined as Pave = ave=1V(Q)Q(x,y,z)dV=mV(Q) , where V(Q) and m are the volume and the mass of Q. respectively. If the density of tile unit ball centered at tile origin is (x,y,z)=ex2y2z2use a CAS to find its average density. Round your answer to three decimal places.Show that the moments of inertia Ix,Iy. and Izabout the yz-plane. x-p1ane. and xy-plane. respectively, of the unit ball centered at the origin whose density is p(x,y,z)=ex2y2z2 are the same. Round your answer to two decimal places.In the following exercises, the function T:SR,T(u,v)=(x,y) on the region S={(u,v)0u1,0v1} bounded by the unit square is given, where RR2 is the image of S under a. Justify that the function T is a C transformation. b. Find the images of the vertices of the unit square S through the function T. c. Determine the image R of the unit square S and graph it. 356. x=2u,y=3v In the following exercises, the function T:SR,T(u,v)=(x,y) on the region S={(u,v)0u1,0v1} bounded by the unit square is given, where RR2 is the image of S under a. Justify that the function T is a C transformation. b. Find the images of the vertices of the unit square S through the function T. c. Determine the image R of the unit square S and graph it. 357. x=u2,yy3 In the following exercises, the function T:SR,T(u,v)=(x,y) on the region S={(u,v)0u1,0v1} bounded by the unit square is given, where RR2 is the image of S under a. Justify that the function T is a C transformation. b. Find the images of the vertices of the unit square S through the function T. c. Determine the image R of the unit square S and graph it. 358. x=uv,y=u+v In the following exercises, the function T:SR,T(u,v)=(x,y) on the region S={(u,v)0u1,0v1} bounded by the unit square is given, where RR2 is the image of S under a. Justify that the function T is a C transformation. b. Find the images of the vertices of the unit square S through the function T. c. Determine the image R of the unit square S and graph it. 359. x=2uv,y=u+2v In the following exercises, the function T:SR,T(u,v)=(x,y) on the region S={(u,v)0u1,0v1} bounded by the unit square is given, where RR2 is the image of S under a. Justify that the function T is a C transformation. b. Find the images of the vertices of the unit square S through the function T. c. Determine the image R of the unit square S and graph it. 360. x=u3,u=v2 In the following exercises, the function T:SR,T(u,v)=(x,y) on the region S={(u,v)0u1,0v1} bounded by the unit square is given, where RR2 is the image of S under a. Justify that the function T is a C1transformation. b. Find the images of the vertices of the unit square S through the function T. c. Determine the image R of the unit square S and graph it. 356. x=2u,y=3v In the following exercises, determine whether transformations T:SRare one-to—one or not. 362. x=u2,y=u2+v , where S is the triangle of (1,0),(1,0),(1,1) , and (0,2)In the following exercises, determine whether transformations T:SRare one-to—one or not. 363. x=u4,y=u4+v , where S is the triangle of vertices (—2, 0). (2. 0). and (0, 2).In the following exercises, determine whether transformations T:SRare one-to—one or not. 364. x=u4,y=3v. where S is the square of vertices (—1, 1), (—1. —1), (1. —1), and (1, 1).In the following exercises, determine whether transformations T:SRare one-to—one or not. 365. T(u,v)=(2uv,u) where S is the triangle of vertices (—1, 1), (—1, —1), and (1, —1).In the following exercises, determine whether transformations T:SRare one-to—one or not. 366. x=u+v+w,y=u+v,z=w, where S=R=R3 In the following exercises, determine whether transformations T:SRare one-to—one or not. 367. x=u2+v+w,y=u2+v,=w where S=R=R3 In the following exercises, the transformations T:SRare one—to—one. Find their related inverse transformations T1:RS 368. x=4u,y=5v,whereS=R=R2 In the following exercises, the transformations T:SRare one—to—one. Find their related inverse transformations T1:RS 369. x=u+2v,y=u+v where S=R=R2 In the following exercises, the transformations T:SRare one—to—one. Find their related inverse transformations T1:RS 370. x=e2u+v,y=euv where S=R2 and R={(x,y)u0,y0}In the following exercises, the transformations T:SRare one—to—one. Find their related inverse transformations T1:RS 271. x=lnu,y=ln(uv), where S={(u,v)u0,v0} and R=R2 .In the following exercises, the transformations T:SRare one—to—one. Find their related inverse transformations T1:RS 372. x=u+v+w,y=3v,z=3w , where S=R=R3 In the following exercises, the transformations T:SRare one—to—one. Find their related inverse transformations T1:RS 373. x=u+v,y=v+w,z=u+w, where S=R=R3 In the following exercises, the transformation T:SR,T(u,v)=(x,y) and the region RR2 are given. Find the region SR2 . 374. x=au,y=bv,R={(x,y)x2+y2a2b2} where a,b0 In the following exercises, the transformation T:SR,T(u,v)=(x,y) and the region RR2 are given. Find the region SR2 . 375. x=au,y=bv,R={(x,y)x2a2+y2b21} where a,b0 In the following exercises, the transformation T:SR,T(u,v)=(x,y) and the region RR2 are given. Find the region SR2 . 376. x=ua,yyb,z=wc R={(x,y)x2+y2+z21} where a,b,c0 In the following exercises, the transformation T:SR,T(u,v)=(x,y) and the region RR2 are given. Find the region SR2 . 377. x=au,y=bv,z=cw,R={(x,y)x2a2y2b2z2c21,z0} where a,b,c0 In the following exercises, find the Jacobian J of the transformation. 378. x=u+2v,y=u+v In the following exercises, find the Jacobian J of the transformation. 379. x=u32,y=vu2 In the following exercises, find the Jacobian J of the transformation. 380. x=e2uv,y=eu+v In the following exercises, find the Jacobian J of the transformation. 381. x=uev,y=ev In the following exercises, find the Jacobian J of the transformation. 382. x=ucos(ev),y=usin(ev)In the following exercises, find the Jacobian J of the transformation. 383. x=vsin(u2),y=vcos(u2)In the following exercises, find the Jacobian J of the transformation. 384. x=ucos(ev),y=vcos(u2)In the following exercises, find the Jacobian J of the transformation. 385. x=vcosh(1u)y=vsinh(1u),z=u+w2 In the following exercises, find the Jacobian J of the transformation. 386. x=u+v,y=v+w,z=u In the following exercises, find the Jacobian J of the transformation. 387. x=uv,y=u+v,z=u+v+w The triangular region R with the vertices (O.O),(1. 1)and(1,2) is shown in the following figure.The triangular region R with the vertices (0, 0). (2, 0), and (1. 3) is shown in the following figure. Find a transformation T:SR . T(u,v)=(x,y)=(au+bv,cu,dv)where a. b. c and d are real numbers with ab0 such that T1(0,0)=(0,0),T1(2,0)=(1,0) , and T1(1,3)=(0,1) . b. Use the transformation T to find the area A(R) of the region R.In the following exercises, use the transformation u=yx,v=y . to evaluate the integrals on the parallelogram R of vertices (0. 0), (1. 0), (2, 1), and(1. 1) shown in the following figure. 390. R(yx)dA In the following exercises, use the transformation u=yx,v=y . to evaluate the integrals on the parallelogram R of vertices (0. 0), (1. 0), (2, 1), and(1. 1) shown in the following figure. 391. R(y2xy)dA In the following exercises, use the transformation yx=u,x+y=v to evaluate the integrals on the lines y=x,y=x+2,y=x+2 . and y=x shown in the following figure. 392. Rex+ydA In the following exercises, use the transformation yx=u,x+y=v to evaluate the integrals on the lines y=x,y=x+2,y=x+2 . and y=x shown in the following figure. 393 Rsin(xy)dA In the following exercises, use the transformation x = U, 5y = v to evaluate the integrals on the region R bounded by the ellipse x2+25y2=1 shown in the following figure. 394. Rx2+25y2dA In the following exercises, use the transformation x = U, 5y = v to evaluate the integrals on the region R bounded by the ellipse x2+25y2=1 shown in the following figure. 395. R( x 2 +25 y 2 )adA In the following exercises, use the transformation u = x + y. V = x — y to evaluate the integrals on the trapezoidal region R determined by the points (1, 0). (2. 0). (0. 2). and (0. 1) shown in the following figure. 396. R(x22x+y2)ex+ydA In the following exercises, use the transformation u = x + y. V = x — y to evaluate the integrals on the trapezoidal region R determined by the points (1, 0). (2. 0). (0. 2). and (0. 1) shown in the following figure. 397. R(x22xy+y2)ex+ydA The circular annulus sector R bounded by the circles 4x2+4y2=1 and 9x2+9y2=64 . the line x=y3. and the y-axis is shown in the following figure. Find a transformation T from a rectangular region S in the r-plane to the region R in the xy-plane. Graph S.The solid R bounded by the circular cylinder x2+y2=9 and the planes z =0, z = 1. x = 0. and y = 0 is shown in the following figure. Find a transformation T from a cylindrical box S in rz-space to the solid R in xyz-space.Show that Rf( x 2 3 + y 2 3 )dA=21501f()dp. where f is a continuous function on LO. 11 and R is the region bounded by the ellipse 5x2+3y2=15 .Show that Rf( 16 x 2 +4y+ x 2 )dv=201f()2dp. where f is a continuous function on 10, 11 and R is the region bounded by the ellipsoid 16x2+4y2+z2=1 .[T] Find the area of the region bounded by the curves xv = 1. xv = 3. y = 2x. and y = 3x by using the transformation u = xy and v=yx Use a computer algebra system (CAS) to graph the boundary curves of the region R.[T] Find the area of the region bounded by the curves x2y=2,x2y=3,y=x,and y = 2x by using the transformation u = x2y and x=yx . Use a CAS to graph the boundary curves of the region R.Evaluate the triple integral 0f11f2zfz+1(y+1)dxdydz by using the transformation u = x —zz, = 3y. and w=z2 .Evaluate the triple integral 4f24f63zf3z+2(54x)dxdzdyby using the transformation u=x—3z. v=4y. and w=z.A transformation T:R2R2,T(u,v)=(x,y)of the form x = au + by. y = cu + dy. where a, b, c, and d are real numbers, is called linear. Show that a linear transformation for which adbc0 maps parallelograms to parallelograms.The transformation T:R2T(u,v)=(x,y) . where x=ucosvsin , y=usin+vcos. is called a rotation of angle . Show that the inverse transformation of Tsatisfies T1=T. where Tis the rotation of angle — .[T] Find the region S in the uv-plane whose image through a rotation of angle is the region R enclosed by the ellipse x2+ 4y2 = 1. Use a CAS to answer the following questions. a. Graph the region S. b. Evaluate the integral se2uvdu dv. Round your answer to two decimal places.[T] The transformations T : R P. i = 1,.... 4. defined by T1(u.v) = (u. —v). T2, (u, v) = (—u. v). T3 (u. v) = (—u. —v), and T4(u. v) = (v. u) are called reflections about the x-axis. y-axis. origin, and the line y = x. respectively. a. Find the image of the region S = {(u. v)u2+ — v2— 2u -4v +1 0} in the xy- plane through the transformation T1T2T3T4b. Use a CAS to graph R. c. Evaluate the Integral Ssin(u2)dudvby using a CAS. Round your answer to two decimal places.[T] The transformation Tk,1,1:R3R3,Tk,1,1(u,v,w)=(x,y,z) of the form x = ku, y = v, z = w. where k1 is a positive real number is called a stretch if k >1 and a compression if 0 < k < 1 in the x-direclion. Use a CAS to evaluate the integral se( 4 x 2 +9 y 2 +25 z 2 )dxdydzon the solid S={(s,y,z)4x2+9y2+25z21} by considering the compression T2,3,5(u,v,w)=(x,y,z)defined by x=u2,y=v3 and z=w5 Round your answer to four decimal places.[T] Find transformations Ta,0:R2R2,Ta,0(u,v)=(u+av+v) , where a0 is a real number, is called a shear in the x-direction. The transformation , Tb,0:R2R2,To,b(u,v)=(u,bu+v) , where b0 is a real number, is calles a shear in the y-direction a. Find transformation T0,2T3,0 b. Find the images R of the trapezoidal region bounded by u=0,v=0, v=1, and v=2-u through the transformation T0,2T3,0c. Use a CAS to graph the image R in the d. Find the area of the region R by using region S.Use the transformation, x=au,y=av,z=cw and spherical coordinates to show that the volume of a legion bounded by the spheroid x2+y2a2+z2c2=1 is 4a2c3 .Find the volume of a football whose shape is a spheroid , x2+y2a2+z2c2=1 whose length from tip to tip is 11 inches and circumference at the center is 22 inches. Round your answer to two decimal places.[T] Lamé ovals (or superellipses) are plane curves of equations (xa)n+(yb)n=1 where a, b, and n are positive real numbers. a. Use a CAS to graph the regions R bounded by Lamé ovals for a= 1.b=2,n=4 and n=6. respectively. b. Find the transformations that map the region R bounded by the Lamé oval x4+y4= 1, also called a squircle and graphed in the following figure, into the unit disk. c. Use a CAS to find an approximation of the area A(R) of the region R x4+y4=1 . Round your answer to two decimal places.[T] Lamé ovals have been consistently used by designers and architects. For instance, Gerald Robinson, a Canadian architect, has designed a parking garage in a shopping center in Peterborough, Ontario, in the shape of a superellipse of the equation (xa)n+(yb)n=1 with ab=97 - and n = e. Use a CAS to find an approximation of the area of the parking garage in the case a = 900 yards, b = 700 yards, and n = 2.72 yards.True or False? Justify your answer with a proof or a counterexample. 416. abcdf(x,y)dydx=cdabf( x,y)dydx True or False? Justify your answer with a proof or a counterexample. 417. Fubini’s theorem can be extended to three dimensions, as long as f is continuous in all variables.True or False? Justify your answer with a proof or a counterexample. 418. The integral 0201r1dzdrdrepresents the volume oor of a tight cone.True or False? Justify your answer with a proof or a counterexample. 419. The Jacobian of the transformation for x=u22v,y=3v2uv is given by 4u26u+4v . Evaluate the following integrals.True or False? Justify your answer with a proof or a counterexample. 420. R(5x3y2y2)dA,R={(x,y)0x2,1y4}True or False? Justify your answer with a proof or a counterexample. 421 Dy3x2+1dA,D={(x,y)0x1,xyx}True or False? Justify your answer with a proof or a counterexample. 422. Dsin(x2+y2)dA where D is a disk of radius 2 centered at the origin True or False? Justify your answer with a proof or a counterexample. 423. 01y1xye x 2dxdy True or False? Justify your answer with a proof or a counterexample. 424. 110z0 xz6dydxdz True or False? Justify your answer with a proof or a counterexample. 425. R3ydV, where R={(x,y,z)0x1,0yx,0z9y2}True or False? Justify your answer with a proof or a counterexample. 426. 0202r1rdzddr True or False? Justify your answer with a proof or a counterexample. 427. 020/213 2sin( )dpd True or False? Justify your answer with a proof or a counterexample. 428. 01 1 x 2 1 x 2 1 x 2 y 2 1 x 2 y 2 dzdydx For the following problems, find the specified area or volume. 429. The area of region enclosed by one petal of r = cos(4 ).For the following problems, find the specified area or volume. 430. The volume of the solid that lies between the paraboloid z=2x2+2y2 and the plane z = 8.For the following problems, find the specified area or volume. 431. The volume of the solid bounded by the cylinder x2+y2=16 and from z= 1 to z+x= 2.For the following problems, find the specified area or volume. 432. The volume of the intersection between two spheres of radius 1, the top whose center is (0. 0. 0.25) and the bottom, which is centered at (0. 0. 0).For the following problems, find the center of mass of the region. 43. (x,y)=(y+1)x on the region bounded by y=ex,y=0, and x=1 .For the following problems, find the center of mass of the region. 434. (x,y)=z on the inverted cone with radius 2 and height 2.For the following problems, find the center of mass of the region. 434. (x,y,z)=z in the region bounded by y=ex,y=0 . and x=1.For the following problems, find the center of mass of the region. 436. The volume an ice cream cone that is given by the solid above z=(x2+y2) and below z2+x2+y2=z.The following problems examine Mount Holly in the state of Michigan. Mount Holly is a landfill that was converted into a ski resort. The shape of Mount Holly can be approximated by a tight circular cone of height 1100 ft and radius 6000 ft. 437. If the compacted trash used to build Mount Holly on average has a density 400 lb/ft3. find the amount of work required to build the mountain.The following problems examine Mount Holly in the state of Michigan. Mount Holly is a landfill that was converted into a ski resort. The shape of Mount Holly can be approximated by a tight circular cone of height 1100 ft and radius 6000 ft. 438. In reality, it is very likely that the trash at the bottom of Mount Holly has become more compacted with all the weight of the above trash. Consider a density function with respect to height: the density at the top of the mountain is still density 400 lb/ft3 and the density increases. Every 100 feet deeper, the density doubles. What is the total weight of Mount Holly?The following problems consider the temperature and density of Eaith’s layers. 439. [T] The temperature of Earth’s layers is exhibited in the table below. Use your calculator to fit a polynomial of degree 3 to the temperature along the radius of the Earth. Then find the average temperature of Earth. (Hint: begin at 0 in the inner core and increase outward toward the surface) Layer Depth from center (km) Temperature °C Rocky Crust 0 to 40 0 Upper Mantle 40 to 150 870 Mantle 400 to 650 870 Inner Mantel 650 to 2700 870 Molten Outer Core 2890 to 5150 4300 Inner Core 5150 to 6378 200 Source: http://www.enchantedlearning.com/siibjects/ astronomy/planets/earth/I nside.shtml [T] The density of Earth’s layers is displayed in the table below. Using your calculator or a computer program, find the best—fit quadratic equation to the density. Using this equation, find the total mass of Earth. Layer Depth from center (kill) Density (g/ cm3) Inner Core 0 1 2.95 Outer Core 1228 11.05 Mantle 3488 5.(X Upper Mantle 6338 3.90 Crust 6378 2.55 Source: http://hyperphysics. phy-astr.gsu.edu/hbase/ geophys/earthstruct. html The following problems concern the Theorem of Pappus (see Moments and Centers of Mass (http:Ilcnx.orgl contentlm53649llatestl) for a refresher), a method for calculating volume using centroids. Assuming a region R, when you revolve around the x-axis the volume is given by Vx=2Ay . and when you revolve around the y-axis the volume is given by Vy=2Ax. where A is the area of R. Consider the legion bounded by x2+y2=1 and above y=x+1 . 441. Find the volume when you revolve the region around the x-axis.The following problems concern the Theorem of Pappus (see Moments and Centers of Mass (http:Ilcnx.orgl contentlm53649llatestl) for a refresher), a method for calculating volume using centroids. Assuming a region R, when you revolve around the x-axis the volume is given by Vx=2Ay . and when you revolve around the y-axis the volume is given by Vy=2Ax. where A is the area of R. Consider the legion bounded by x2+y2=1 and above y=x+1 . 442. Find the volume when you revolve the region around the y-axis.The domain of vector field F = F(x, y) is a set of points (x, y) in a plane, and the range of F is a set of what in the plane?For the following exercises, determine whether the statement is true or false. 2. Vector field F=3x2,11(x,y)=x3+y and 2(x,y)=y+x3+100.For the following exercises, determine whether the statement is true or false. 3. Vector field F=y,xx2+y2 is constant in direction and magnitude on a unit circle.For the following exercises, determine whether the statement is true or false. 4. Vector field F=y,xx2+y2 is neither a radial field nor a rotation.For the following exercises, describe each vector field by drawing some of its vectors. 5. [T]F(x,y)=xi+yj For the following exercises, describe each vector field by drawing some of its vectors. 6. [T]F(x,y)=yi+xj For the following exercises, describe each vector field by drawing some of its vectors. 7. [T]F(x,y)=xiyj For the following exercises, describe each vector field by drawing some of its vectors. 8. [T]F(x,y)=i+j For the following exercises, describe each vector field by drawing some of its vectors. 9. [T]F(x,y)=2xi+3yj For the following exercises, describe each vector field by drawing some of its vectors. 10. [T]F(x,y)=3i+xj For the following exercises, describe each vector field by drawing some of its vectors. 11. [T]F(x,y)=yi+sinxj For the following exercises, describe each vector field by drawing some of its vectors. 12. [T]F(x,y,z)=xi+yj+zk For the following exercises, describe each vector field by drawing some of its vectors. 13. [T]F(x,y,z)=2xi2yj2zk For the following exercises, describe each vector field by drawing some of its vectors. 14. [T]F(x,y,z)=yzixzj For the following exercises, find the gradient vector field of each function f. 15.f(x,y)=xsiny+cosy For the following exercises, find the gradient vector field of each function f. 16.f(x,y,z)=zexy For the following exercises, find the gradient vector field of each function f. 17.f(x,y,z)=x2y+xy+y2z For the following exercises, find the gradient vector field of each function f. 18.f(x,y)=x2sin(5y)For the following exercises, find the gradient vector field of each function f. 19.f(x,y)=In(1+x2+2y2)For the following exercises, find the gradient vector field of each function f. 20.f(x,y,z)=xcos(yz)What is vector field F(x, y) with a value at (x, y) that is of unit length and points toward (1, 0)?For the following exercises, write formulas for the vector fields with the given properties. 22. All vectors are parallel to the x-axis and all vectors on a vertical line have the same magnitude.For the following exercises, write formulas for the vector fields with the given properties. 24. All vectors are of unit length and are peipendicular to the position vector at that point.For the following exercises, write formulas for the vector fields with the given properties. 23. All vectors point toward the origin and have constant length.Give a formula F(x, y) = M(x, y)i + N(x, y)j for the vector field in a plane that has the properties that F = 0 at (0, 0) and that at any other point (a, b). F is tangent to circle x2+ y2 = a2 + b2and points in the clockwise direction with magnitude |F|=a2+b2.Is vector field F(x, y) = (P(x, y), Q(x, y)) = (sin x + y)i + (cos y + x)j a gradient field?Find a formula for vector field F(x, y) = M(x,,y)i + N(x, y)j given the fact that for all points (x, y), F points toward the origin and |F|=10x2+y2 For the following exercises, assume that an electric field in the xy-plane caused by an infinite line of charge along the x-axis is a gradient field with potential function V(x, y) = c In (r0 x 2 + y 2 ), where c > 0 is a constant and r0is a reference distance at which the potential is assumed to be zero. 29. Show that the electric field at a point in the xy-plane is directed outward from the origin and has magnitude El = cr ,. where r = x2=y2 A flow line (or sweamline) of a vector field F is a curve r(t) such that dr/dt = F(r(t)). If F represents the velocity field of a moving particle, then the flow lines are paths taken by the particle. Therefore, flow lines are tangent to the vector field. For the following exercises, show that the given curve c(t) is a flow line of the given velocity vector field F(x, y,z).For the following exercises, assume that an electric field in the xy-plane caused by an infinite line of charge along the x-axis is a gradient field with potential function V(x, y) = c In (r0 x 2 + y 2 ), where c > 0 is a constant and r0is a reference distance at which the potential is assumed to be zero.For the following exercises, assume that an electric field in the xy-plane caused by an infinite line of charge along the x-axis is a gradient field with potential function V(x, y) = c In (r0 x 2 + y 2 ), where is a constant and r0is a reference distance at which the potential is assumed to be zero. 28. Find the components of the electric field in the x— and y—directions, where E(x, y) = -v V(x. v). 29. Show that the electhc field at a point in the xy-plane is directed outward from the origin and has magnitude El = cr -. where r = x2=y2 A flow line (or sweamline) of a vector field F is a curve r(t) such that dr/dt = F(r(t)). If F represents the velocity field of a moving particle, then the flow lines are paths taken by the particle. Therefore, flow lines are tangent to the vector field. For the following exercises, show that the given curve c(t) is a flow line of the given velocity vector field F(x, y, z).c(t) = (sin t. cos t, et); F(x,y,z)=y,x,z For the following exercises, let F = xi + yi, G = -yi + xj. and H = xi-yj. Match F, G, and H with their graphs.For the following exercises, let F = xi + yi, G = -yi + xj. and H = xi-yj. Match F, G, and H with their graphs.For the following exercises, let F = xi + yi, G = -yi + xj. and H = xi-yj. Match F, G, and H with their graphs.For the following exercises, let F = xi + yj, G = -yi + xj, andH = -xj + yj. Match the vector fields with their graphs in (I)- (IV). a. F+G b. F+H C. G+H d. -F+G For the following exercises, F=xi+yj,G=yi+xj,H=xi+yj vector fields with their graphs in (I)- (IV). a. F+G b. F+H C. G+H d. -F+G For the following exercises, let F=xi+yj,G=yi+xj,H=xi+yj . Match the vector fields with their graphs in (I)- (IV). a. F+G b. F+H C. G+H d. -F+G For the following exercises, let F=xi+yj,G=yi+xj,H=xi+yj . Match the vector fields with their graphs in (I)- (IV). a. F+G b. F+H C. G+H d. -F+G True or False? Line integral cf(x,y)dsis equal to a definite integral if C is a smooth curve defined on [a, b] and if function f is continuous on some region that contains curve C.True or False? Vector functions r1= ti +t2j, 0t1,and r2=(1 —t)i+(1 —t)2j, 0t1, define the same oriented curve.True or False? c(Pdx+Qdy)=c(PdxQdy)True or False? A piecewise smooth cuive C consists of a finite number of smooth cuives that are joined together end to end.True or False?If C is given by x(t) = t,y(t) = t,0 t1. then xvds=fr2d,.For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path. 44. [T] c(x,y)dsC:x=t ,y=(1-t),z=O from (0, 1, 0) to(1, 0, 0)For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path. 45.[T]c(xy)dsC:r(t)=4ti+3tjwhen0t2 For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path. 46.[T] c( x 2 + y 2 + z 2 )dsC:r(t)=sinti+costj+8tkwhen0t2 For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path. 44.[T]c(x+y)dsC:x=t,y=(1t),z=0from(0,1,0)to(1,0,0)For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path. 45. [T]c(xy)dsC:r(t)=4ti+3tjwhen 0t2 For the following exercises, find the work done. 49. Find the work done by vector field F(x,y,z)=xi+3xyj(x+z)k on a particle moving along a line segment that goes from (1, 4, 2) to (0, 5, 1).For the following exercises, find the work done. 50. Find the work done by a person weighing 150 lb walking exactly one revolution tip a circular, spiral staircase of radius 3 ft if the person rises 10 ft.For the following exercises, find the work done. 51. Find the work done by force field F(x,y,z)=12xi12yj+12k on a particle as it moves along the helix r(t)=costi+sintj+tkfrom point (1, 0, 0) to point (-1,0, 3).For the following exercises, find the work done. 52. Find the work done by vector field F(x,y,)=yi+2xj in moving an object along path C, which joins points (1, 0) and (0, 1).For the following exercises, find the work done. 53. Find the work done by force F(x,y,)=2yi+3xj+(x+y)k in moving an object along curve r(t) = cos(t)i + sin(t)j + 16k. where 0t2 For the following exercises, find the work done. 54. Find the mass of a wire in the shape of a circle of radius 2 centered at (3, 4) with linear mass density p(x, y) = y2.For the following exercises, evaluate the line integrals. 55. Evaluate cF.dr,where F(x,y)=1j, and C is the part of the graph of y=12x3x from (2, 2) to (-2, -2).For the following exercises, evaluate the line integrals. 56. Evaluate c( x 2 + y 2 + z 2 )1ds,where is the helix x= cost, y= sint, z= t (0tT).For the following exercises, evaluate the line integrals. 57. Evaluate cyzdx+xzdy+xydzover the line segment from (1, 1, 1) to (3, 2, 0).For the following exercises, evaluate the line integrals. 58. Let C be the line segment from point (0, 1, 1) to point (2, 2, 3). Evaluate line integral cyds.For the following exercises, evaluate the line integrals. 59. [T] Use a computer algebra system to evaluate the line integral cy2dx+xdy,where C is the arc of the parabola x = 4-x2from (-5, -3) to (0, 2).For the following exercises, evaluate the line integrals. 60. [T] Use a computer algebra system to evaluate the line integral c(x+3 y 2)dy,over the path C given by x = 2t, y = 10t where (0tT).For the following exercises, evaluate the line integrals. 61. [T] Use a CAS to evaluate line integral cxydx+ydyover path C given by x = 2t, y = 10t, where 0t1 For the following exercises, evaluate the line integrals. 62. Evaluate line integral c(2xy)dx+(x+3y)dy, where C lies along the x-axis from x = 0 to x = 5.For the following exercises, evaluate the line integrals 63. [T] Use a CAS to evaluate yc2 x 2 y 2ds, where C is x = t,y = t, 0t5.For the following exercises, evaluate the line integrals 64. [T] Use a CAS to evaluate cxyds,where C is x=t2,y=4t,0t1 .In the following exercises, find the work done by force fiel d F on an object moving along the indicated path. 65. F(x,y)=xi2yjC: y=x3 from (0, 0) to (2, 8)In the following exercises, find the work done by force fiel d F on an object moving along the indicated path. 66. F(x,y)=2xi+yjC: counterclockwise around the triangle with vertices (0, 0), (1, 0), and (1, 1)In the following exercises, find the work done by force fiel d F on an object moving along the indicated path. 67. F(x,y,z)=xi+yj5zkC: r(t)=2costi+2sintj+tk,0t2 In the following exercises, find the work done by force field F on an object moving along the indicated path. 68. Let F be vector field F(x,y,)=(y2+2xey+1)i+(2xy+x2ey+2y)j. Compute the work of integral cF.dr,where C is the path r(t)=sinti+costj,0t2 .In the following exercises, find the work done by force fiel d F on an object moving along the indicated path. 69. Compute the work done by force F(x,y,z)=2xi+3yjzk along path r(t)=ti+t2j+t3k,where0t1 In the following exercises, find the work done by force fiel d F on an object moving along the indicated path. 70. Evaluate cF.dr,where F(x,y,)=1x+yi+1x+yj and C is the segment of the unit circle going counterclockwise from (1, 0) to (0, 1).In the following exercises, find the work done by force fiel d F on an object moving along the indicated path. 71. Force F(x,y,z)=zyi+xj+z2xkacts on a particle that travels from the origin to point (1, 2, 3). Calculate the work done if the particle travels:a. along the path (0,0,0) .(1,0,0) ’(1,2,0) .(1,2,3) along straight-line segments joining each pair of endpoints: b. along the straight line joining the initial and final points. c. Is the work the same along the two paths? (1, 2, 3)In the following exercises, find the work done by force fiel d F on an object moving along the indicated path. 72. Find the work done by vector field F(x,y,z)=xi+3xyj(x+z)k on a particle moving along a line segment that goes from (1, 4, 2) to (0, 5, 1).In the following exercises, find the work done by force fiel d F on an object moving along the indicated path. 73. How much work is required to move an object in vector field F(x,y,)=yi+3xj along the upper part of ellipse x24+y2=1 from (2, 0) to (-2, 0)?In the following exercises, find the work done by force fiel d F on an object moving along the indicated path. 74.A vector field is given F(x,y,)=(2x+3y)i+(3x+2y)j . Evaluate the integral of the field around a circle of unit radius traversed in a clockwise fashion.Evaluate the line integral of scalar function xy along parabolic path y = x2connecting the origin to point (1,1).Find yc2dx+(xy x 2)dy along C: y = 3x from C (0, 0) to (1, 3).Find yc2dx+(xy x 2)dyalong C: y2= 9x from (0, 0) to (1, 3).For the following exercises, use a CAS to evaluate the given line integrals. 78. [T] Evaluate F(x,y,z)=x2zi+6yj+yz2k, where C is represented by r(t)=ti+t2j+Intk,1t3 For the following exercises, use a CAS to evaluate the given line integrals. 79. [T] Evaluate line integral yxeydswhere, is the arc of curve x=ey from (1, 0) to (e, 1).For the following exercises, use a CAS to evaluate the given line integrals. 80. [T] Evaluate the integral xy2ds,where triangle with vertices (0, 1, 2), (1, 0, 3), and (0, -1, 0).For the following exercises, use a CAS to evaluate the given line integrals. 81. [T] Evaluate line integral xy2ds,where y is Curve y = in x from (1, 0) toward (e, 1).For the following exercises, use a CAS to evaluate the given line integrals. 82. [T] Evaluate line integral xy4ds,where is the right half of circle x2+y2=16 .For the following exercises, use a CAS to evaluate the given line integrals. 83. [T] Evaluate cF.dr,where F(x,y,z)=x2yi+(xz)j+xyzk and C: r(t)=ti+t2j+2k,0t1 .For the following exercises, use a CAS to evaluate the given line integrals. 84. Evaluate cF.dr, where F(x,y,)=2xsin(y)i+(x2cos(y)3y2)j and C is any path from (-1, 0) to (5, 1).For the following exercises, use a CAS to evaluate the given line integrals. 85. Find the line integral of F(x,y,z)=12x2i5xyj+xzk over path C defined by y = x2, z = x3from point (0, 0, 0) to point (2, 4, 8).For the following exercises, use a CAS to evaluate the given line integrals. 86. Find the line integral of c(1+ x 2y)ds,where C is ellipse r(t)=2costi+3sintjfrom0t.For the following exercises, find the flux. 87. Compute the flux of F = x2i + yj across a line segment from (0, 0) to (1, 2).For the following exercises, find the flux. 88. Let F = 5i and let C be curve y=0,0x4 . Find the flux across C.For the following exercises, find the flux. 89. Let F = 5j and let C be curve y = 0. y=0,0x4 . Find the flux across C.For the following exercises, find the flux. 90. Let F = -yi + xj and let C: r(t) = cos t + sin tj ( 0t2 ). Calculate the flux across C.For the following exercises, find the flux. 91. Let F = (x2+y2)i + (2xy)j. Calculate flux F orientated counterclockwise across curve C: x2+ y2= 9.Find the line integral of k c z 2dx+ydy+2ydz,where C consists of two parts: C1 and C2. C1 is the intersection of cylinder x2+ y2 = 16 and plane z = 3 from (0, 4, 3) to (-4, 0, 3). C2is a line segment from (-4, 0, 3) to (0, 1, 5).A spring is made of a thin wire twisted into the shape of a circular helix x = 2 cos t, y = 2 sin t. z = t. Find the mass of two turns of the spring if the wire has constant mass density.A thin wire is bent into the shape of a semicircle of radius a. If the linear mass density at point P is directly proportional to its distance from the line through the endpoints, find the mass of the wire.An object moves in force field F(x,y,z)=y2i+2(x+1)yj counterclockwise from point (2, 0) along elliptical path x2+ 4y2= 4 to (-2, 0), and back to point (2, 0) along the x-axis. How much work is done by the force field on the object?Find the work done when an object moves in force field F(x, y ,z) = 2xi — (x + z)j + (y — x)k along the path given by r(t) = t2i+(t2-t)j+ 3k, 0t1.If an inverse force field F. is given by F(x, y, z) = k r3r, . where k is a constant, find the work done by F as its point of application moves along the x-axis from A(1, 0, 0) to B(2, 0, 0).David and Sandra plan to evaluate line integral cF.dr, along a path in the xy-plane from (0, 0) to (1,1). The force field is F(x,y)=(x+2y)i+(x+y2)j. David chooses the path that runs along the -axis from (0, 0) to (1, 0) and then runs along the vertical line x = 1 from (1, 0) to the final point (1, 1). Sandra chooses the direct path along the diagonal line y = x from (0, 0) to (1, 1). Whose line integral is larger and by how much?True or False? If vector field F is conservative on the open and connected region D, then line integrals of F are path independent on D, regardless of the shape of D.Trueor False? Function r(t) = a + t(b — a), where 0t1, parameterizes the straight-line segment from a to b.True or False? Vector field F(x, y,z) = (y sinz)i + (x sinz)j + (xy cosz)k is conservative.True or False?Vector field F(x,y,z)= yi + (x + z)j — yk is conservative.Justify the Fundamental Theorem of Line Integrals for cF.dr,in the case when F(x, y) = (2x + 2y)i + (2x + 2y)j and C is a portion of the positively oriented circle x2+ y2= 25 from (5, 0) to (3, 4)[T] Find cF.dr,,] where F(x,y)=(yexy+cosx)i+(xexy+1y2+1)j and C is a portion of curve y=sinx from x = 0 to x=2 .[T] Evaluate line integral cF.dr, where F(x,y)=(exsinyy)i+(excosyx2)j, and C is the path given by r(t)=[t3sint2]i[2cos(t2+2)]j for 0t1 .For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function. 106.F(x,y)=2x3i+3y2x2j For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function. 107.F(x,y)=(y+exsiny)i+[(x+2)excosy]j For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function. 108.F(x,y)=(e2xsiny)i+[e2xcosy]j For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function. 109.F(x,y)=(6x+5y)i+(5x+4y)j For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function. 110.F(x,y)=[2xcos(y)ycos(x)]i+[x2sin(y)sin(x)]j For the following exercises, determine whether the vector field is conservative and, if it is, find the potential function. 111.F(x,y)=[yex+sin(y)]i+[ex+xcos(y)]j For the following exercises, evaluate the line integrals using the Fundamental Theorem of Line Integrals. 112.c(yi+xj).dr,where C is any path from (0, 0) to (2,4)For the following exercises, evaluate the line integrals using the Fundamental Theorem of Line Integrals. 113.c(2ydx+2xdy), where C is line segment from (0, 0) to (4,4)For the following exercises, evaluate the line integrals using the Fundamental Theorem of Line Integrals. 114.[T]c[arctanyx xy x 2 + y 2 ]dx+[ x 2 x 2 + y 2 +e y( 1y)]dy, where C is any smooth curve from (1, 1) to (-1,2)For the following exercises, evaluate the line integrals using the Fundamental Theorem of Line Integrals. 115. Find the conservative vector field for the potential function f(x,y)=5x2+3xy+10y2.For the following exercises, determine whether the vector field is conservative and, if so, find a potential function. 117.F(x,y)=(excosy)i+6(exsiny)j For the following exercises, determine whether the vector field is conservative and, if so, find a potential function. 118.F(x,y)=(2xyzex2y)i+6(x2ex2y)j For the following exercises, determine whether the vector field is conservative and, if so, find a potential function. 119.F(x,y,z)=(yez)i+(xez)j+(xyez)k For the following exercises, determine whether the vector field is conservative and, if so, find a potential function. 120.F(x,y,z)=(siny)i(xcosy)j+k For the following exercises, determine whether the vector field is conservative and, if so, find a potential function. 116.F(x,y)=(12xy)i+6(x2+y2)j For the following exercises, determine whether the vector field is conservative and, if so, find a potential function. 121.F(x,y,z)=(1y)i+(xy2)j+(2z1)k For the following exercises, determine whether the vector field is conservative and, if so, find a potential function. 122.F(x,y,z)=3z2i-cosyj+2xzk For the following exercises, determine whether the vector field is conservative and, if so, find a potential function. 123.F(x,y,z)=(2xy)i+(x2+2yz)j+y2k For the following exercises, determine whether the given vector field is conservative and find a potential function. 124.F(x,y)=(excosy)i+6(exsiny)j For the following exercises, determine whether the given vector field is conservative and find a potential function. 125.F(x,y)=(2xyex2y)i+6(x2ex2y)j For the following exercises, evaluate the integral using the Fundamental Theorem of Line Integrals. 126. Evaluate cf.dr, where f(x,y,z)=cos(x)+sin(y)xyzand C is any path that starts at (1,12,2) and ends at (2, 1, -1).For the following exercises, evaluate the integral using the Fundamental Theorem of Line Integrals. 127. [T] Evaluatecf.dr, where f(x, y) = xy +exand C is a straight line from (0, 0) to (2, 1).For the following exercises, evaluate the integral using the Fundamental Theorem of Line Integrals. 128. [T] Evaluate cf.dr, where f(x, y) = x2y- x and C is any path in a plane from (1, 2) to(3,2).For the following exercises, evaluate the integral using the Fundamental Theorem of Line Integrals. 129. Evaluate cf.dr,where f(x,y,z)=xyz2yz and C has initial point (1, 2) and terminal point (3, 5).For the following exercises, let F(x, y) = 2xy2i + (2yx2+2y)j and G(x, y) = (y + x)i + (y — x)j, and let C1be the curve consisting of the circle of radius 2, centered at the origin and oriented counterclockwise, and C2be the curve consisting of a line segment from (0, 0) to (1, 1) followed by a line segment from (1, 1) to (3, 1). 130. Calculate the line integral of F over Ci.For the following exercises, let F(x, y) = 2xy2i + (2yx2+2y)j and G(x, y) = (y + x)i + (y — x)j, and let C1be the curve consisting of the circle of radius 2, centered at the origin and oriented counterclockwise, and C2be the curve consisting of a line segment from (0, 0) to (1, 1) followed by a line segment from (1, 1) to (3, 1). 131. Calculate the line integral of G over C1.For the following exercises, let F(x, y) = 2xy2i + (2yx2+2y)j and G(x, y) = (y + x)i + (y — x)j, and let C1be the curve consisting of the circle of radius 2, centered at the origin and oriented counterclockwise, and C2be the curve consisting of a line segment from (0, 0) to (1, 1) followed by a line segment from (1, 1) to (3, 1). 132. Calculate the line integral of F over C2.For the following exercises, let F(x, y) = 2xy2i + (2yx2+2y)j and G(x, y) = (y + x)i + (y — x)j, and let C1be the curve consisting of the circle of radius 2, centered at the origin and oriented counterclockwise, and C2be the curve consisting of a line segment from (0, 0) to (1, 1) followed by a line segment from (1, 1) to (3, 1). 133. Calculate the line integral of G over C2.[T] Let F(x, y, z) = x2i + zsin(yz)j + y sin(yz)k. Calculate cF.dr. where C is a path from A = (0,0,1) to B =(3, 1, 2).[T] Find line integral cF.dr,of vector field F(x, y, z) = 3x2z2j + z2 j + (x3+ 2yz)k along curve parameterized by r(t)=(IntIn2)i+t3/2j+tcos(t),1t4.For the following exercises, show that the following vector fields ate conservative by using a computer. Calculate cF.dr for the given curve. 136. F=(xy2+3x2y)i+(x+y)x2j;C is the curve consisting of line segments from (1, 1) to (0, 2) to (3, 0).For the following exercises, show that the following vector fields ate conservative by using a computer. Calculate cF.dr for the given curve. 137.F=2xy2+1i2y(x2+1)(y2+1)2j;C is Parameterized by x=t31,y=t6t,0t1 For the following exercises, show that the following vector fields ate conservative by using a computer. Calculate cF.dr for the given curve. 138. [T] F=[cos(xy2)xy2sin(xy2)]i2x2ysin(xy2)j;C is curve (et,et+1),1t0.For the following exercises, show that the following vector fields ate conservative by using a computer. Calculate cF.dr for the given curve 139. The mass of Earth is approximately 6 × 1027 g and that of the Sun is 330,000 times as much. The gravitational constant is 6.7 × 10-8cm3/s2 g. The distance of Earth from the Sun is about 1 .5 × 1012 cm. Compute, approximately, the work necessary to increase the distance of Earth from the Sun by 1 cm.For the following exercises, show that the following vector fields ate conservative by using a computer. Calculate cF.dr for the given curve 140. [T] Let F = (x, y, z) = (exsiny)i + (excosy)j +z2k. Evaluate the integral cF.ds,where c(t)=(t,t3,et),0t1 For the following exercises, show that the following vector fields ate conservative by using a computer. Calculate cF.dr for the given curve 141. [T] Let C: [1, 2] 2be given by x=et1,y=sin(t). Use a computer to compute the integral cF.ds=c2xcosydxx2sinydy,where F=(2xcosy)i(x2siny)j .For the following exercises, show that the following vector fields ate conservative by using a computer. Calculate cF.dr for the given curve 142. [T] Use a computer algebra system to find the mass of a wire that lies along curve r(t)=(t21)j+2tk,0t1. if the density is 32t .Find the circulation and flux of field F=yi+xj around and across the closed semicircular path that consists of semicircular arch r1(t)=(acost)i+(asint)j,0t. followed by line segment r1(t)=(acost)i+(asint)j,0ta.Compute ccosxcosydxsinxsinydy, where c(t)=(t,t2),0t1.Complete the proof of The Path Independence Test for Conservative Fields by showing that fy= Q(x, y).Measuring Area from a Boundary: The Planimeter Figure 6.47 This magnetic resonance image of a patient’s brain shows a tumor, which is highlighted in red. (credit: modification of wood’ by Christaras A Wikimedia Commons) Imagine you are a doctor who has just received a magnetic resonance image of your patent’s brain. The brain has a tumor (Figure 6.47). How large is the tumor? To be precise, what is the area of the red region? The red cross-section of the tumor has an irregular shape, and therefore it is unlikely that you would be able o find a set of equations or inequalities for the region and then be able to calculate its area by conventional means. You could approximate the area by chopping the region into tiny squares (a Riemann sum approach), but this method always gives an answer with some error. Instead of trying to measure the area of the region directly, we can use a device called a rolling planimeter to calculate the area of the region exactly, simply by measuring its boundary. In this project you investigate how a planimeter works, and you use Green’s theorem to show the device calculates area correctly. A rolling planimeter is a device that measures the area of a planar region by tracing out the boundary of that region (Figure 6.48). To measure the area of a region, we simply run the tracer of the planimeter around the boundary of the region. The planimeter measures the number of turns through which the wheel rotates as we (race the boundary; the area of the shape is proportional to this number of wheel ruins. We can derive the precise proportionality equation using Green’s theorem. As the (racer moves around the boundary of the region, the tracer arm rotates and the roller moves back and forth (but does no; rotate). Figure 6.48 (a) A rolling planimeter. The pivot allows the tracer arm to rotate. The roller itself does not rotate; it only moves back and forth. (b) An interior view of a rolling planimeter. Notice that the wheel cannot rum if the planimeter is moving back and forth th the tracer arm perpendicular to the roller. Let C denote the boundary of region D. the area to be calculated. As the tracer traverses curve C, assume the roller moves along the y-axis (since the roller does not rotate, one can assume it moves along a straight line). Use the coordinates (x, y) to represent points on boundary C, and coordinates (0, y) to represent the position of the pivot. As the planimeter traces C. the pivot moves along the y-axis while the tracer arm rotates on the pivot. Watch a short animation (http://www.openstaxcollege.org/I/20_pianimeter) of a planimeter in action. Begin the analysis by considering the motion of the tracer as it moves from point (x, y) counterclockwise to point (x + dx. y + dy) that is close to (x, y) (Figure 6.49). The pivot also moves, from point (0, y) to nearby point (0, y + dy). How much does the wheel turn as a result of this motion? To answer this question, break the motion into two parts. First, roll the pivot along they-axis from (0. y) to (0, y + dy) without rotating the tracer arm. The tracer arm then ends up a point (x, y + dy) while maintaining a constant angle with the x-axis. Second, rotate the tracer arm by an angle d without moving the roller. Now the tracer is at point (x + dx, y + dy). Let 1 be the distance from the pivot to the wheel and let L be the distance from the pivot to the tracer (the length of the tracer arm). Figure 6.49 Mathematical analysis of the motion of the planimeter. 1. Explain why the total distance through which the wheel rolls the small motion just described is sindy+ld=xLdY+ld.Measuring Area from a Boundary: The Planimeter Figure 6.47 This magnetic resonance image of a patient’s brain shows a tumor, which is highlighted in red. (credit: modification of wood’ by Christaras A Wikimedia Commons) Imagine you are a doctor who has just received a magnetic resonance image of your patent’s brain. The brain has a tumor (Figure 6.47). How large is the tumor? To be precise, what is the area of the red region? The red cross-section of the tumor has an irregular shape, and therefore it is unlikely that you would be able o find a set of equations or inequalities for the region and then be able to calculate its area by conventional means. You could approximate the area by chopping the region into tiny squares (a Riemann sum approach), but this method always gives an answer with some error. Instead of trying to measure the area of the region directly, we can use a device called a rolling planimeter to calculate the area of the region exactly, simply by measuring its boundary. In this project you investigate how a planimeter works, and you use Green’s theorem to show the device calculates area correctly. A rolling planimeter is a device that measures the area of a planar region by tracing out the boundary of that region (Figure 6.48). To measure the area of a region, we simply run the tracer of the planimeter around the boundary of the region. The planimeter measures the number of turns through which the wheel rotates as we (race the boundary; the area of the shape is proportional to this number of wheel ruins. We can derive the precise proportionality equation using Green’s theorem. As the (racer moves around the boundary of the region, the tracer arm rotates and the roller moves back and forth (but does no; rotate). Figure 6.48 (a) A rolling planimeter. The pivot allows the tracer arm to rotate. The roller itself does not rotate; it only moves back and forth. (b) An interior view of a rolling planimeter. Notice that the wheel cannot rum if the planimeter is moving back and forth th the tracer arm perpendicular to the roller. Let C denote the boundary of region D. the area to be calculated. As the tracer traverses curve C, assume the roller moves along the y-axis (since the roller does not rotate, one can assume it moves along a straight line). Use the coordinates (x, y) to represent points on boundary C, and coordinates (0, y) to represent the position of the pivot. As the planimeter traces C. the pivot moves along the y-axis while the tracer arm rotates on the pivot. Watch a short animation (http://www.openstaxcollege.org/I/20_pianimeter) of a planimeter in action. Begin the analysis by considering the motion of the tracer as it moves from point (x, y) counterclockwise to point (x + dx. y + dy) that is close to (x, y) (Figure 6.49). The pivot also moves, from point (0, y) to nearby point (0, y + dy). How much does the wheel turn as a result of this motion? To answer this question, break the motion into two parts. First, roll the pivot along they-axis from (0. y) to (0, y + dy) without rotating the tracer arm. The tracer arm then ends up a point (x, y + dy) while maintaining a constant angle with the x-axis. Second, rotate the tracer arm by an angle d without moving the roller. Now the tracer is at point (x + dx, y + dy). Let 1 be the distance from the pivot to the wheel and let L be the distance from the pivot to the tracer (the length of the tracer arm). Figure 6.49 Mathematical analysis of the motion of the planimeter. 2.showthatcd=0 easuring Area from a Boundary: The Planimeter Figure 6.47 This magnetic resonance image of a patient’s brain shows a tumor, which is highlighted in red. (credit: modification of wood’ by Christaras A Wikimedia Commons) Imagine you are a doctor who has just received a magnetic resonance image of your patent’s brain. The brain has a tumor (Figure 6.47). How large is the tumor? To be precise, what is the area of the red region? The red cross-section of the tumor has an irregular shape, and therefore it is unlikely that you would be able o find a set of equations or inequalities for the region and then be able to calculate its area by conventional means. You could approximate the area by chopping the region into tiny squares (a Riemann sum approach), but this method always gives an answer with some error. Instead of trying to measure the area of the region directly, we can use a device called a rolling planimeter to calculate the area of the region exactly, simply by measuring its boundary. In this project you investigate how a planimeter works, and you use Green’s theorem to show the device calculates area correctly. A rolling planimeter is a device that measures the area of a planar region by tracing out the boundary of that region (Figure 6.48). To measure the area of a region, we simply run the tracer of the planimeter around the boundary of the region. The planimeter measures the number of turns through which the wheel rotates as we (race the boundary; the area of the shape is proportional to this number of wheel ruins. We can derive the precise proportionality equation using Green’s theorem. As the (racer moves around the boundary of the region, the tracer arm rotates and the roller moves back and forth (but does no; rotate). Figure 6.48 (a) A rolling planimeter. The pivot allows the tracer arm to rotate. The roller itself does not rotate; it only moves back and forth. (b) An interior view of a rolling planimeter. Notice that the wheel cannot rum if the planimeter is moving back and forth th the tracer arm perpendicular to the roller. Let C denote the boundary of region D. the area to be calculated. As the tracer traverses curve C, assume the roller moves along the y-axis (since the roller does not rotate, one can assume it moves along a straight line). Use the coordinates (x, y) to represent points on boundary C, and coordinates (0, y) to represent the position of the pivot. As the planimeter traces C. the pivot moves along the y-axis while the tracer arm rotates on the pivot. Watch a short animation (http://www.openstaxcollege.org/I/20_pianimeter) of a planimeter in action. Begin the analysis by considering the motion of the tracer as it moves from point (x, y) counterclockwise to point (x + dx. y + dy) that is close to (x, y) (Figure 6.49). The pivot also moves, from point (0, y) to nearby point (0, y + dy). How much does the wheel turn as a result of this motion? To answer this question, break the motion into two parts. First, roll the pivot along they-axis from (0. y) to (0, y + dy) without rotating the tracer arm. The tracer arm then ends up a point (x, y + dy) while maintaining a constant angle with the x-axis. Second, rotate the tracer arm by an angle d without moving the roller. Now the tracer is at point (x + dx, y + dy). Let 1 be the distance from the pivot to the wheel and let L be the distance from the pivot to the tracer (the length of the tracer arm). Figure 6.49 Mathematical analysis of the motion of the planimeter. 3. Use step 2 to show that the total rolling distance of the wheel as the tracer traverses curve C is Total wheel roll =1LcxdY.Now that you have an equation for the total rolling distance of the wheel, connect this equation to Green’s theorem to calculate area D enclosed by C.Measuring Area from a Boundary: The Planimeter Figure 6.47 This magnetic resonance image of a patient’s brain shows a tumor, which is highlighted in red. (credit: modification of wood’ by Christaras A Wikimedia Commons) Imagine you are a doctor who has just received a magnetic resonance image of your patent’s brain. The brain has a tumor (Figure 6.47). How large is the tumor? To be precise, what is the area of the red region? The red cross-section of the tumor has an irregular shape, and therefore it is unlikely that you would be able o find a set of equations or inequalities for the region and then be able to calculate its area by conventional means. You could approximate the area by chopping the region into tiny squares (a Riemann sum approach), but this method always gives an answer with some error. Instead of trying to measure the area of the region directly, we can use a device called a rolling planimeter to calculate the area of the region exactly, simply by measuring its boundary. In this project you investigate how a planimeter works, and you use Green’s theorem to show the device calculates area correctly. A rolling planimeter is a device that measures the area of a planar region by tracing out the boundary of that region (Figure 6.48). To measure the area of a region, we simply run the tracer of the planimeter around the boundary of the region. The planimeter measures the number of turns through which the wheel rotates as we (race the boundary; the area of the shape is proportional to this number of wheel ruins. We can derive the precise proportionality equation using Green’s theorem. As the (racer moves around the boundary of the region, the tracer arm rotates and the roller moves back and forth (but does no; rotate). Figure 6.48 (a) A rolling planimeter. The pivot allows the tracer arm to rotate. The roller itself does not rotate; it only moves back and forth. (b) An interior view of a rolling planimeter. Notice that the wheel cannot rum if the planimeter is moving back and forth th the tracer arm perpendicular to the roller. Let C denote the boundary of region D. the area to be calculated. As the tracer traverses curve C, assume the roller moves along the y-axis (since the roller does not rotate, one can assume it moves along a straight line). Use the coordinates (x, y) to represent points on boundary C, and coordinates (0, y) to represent the position of the pivot. As the planimeter traces C. the pivot moves along the y-axis while the tracer arm rotates on the pivot. Watch a short animation (http://www.openstaxcollege.org/I/20_pianimeter) of a planimeter in action. Begin the analysis by considering the motion of the tracer as it moves from point (x, y) counterclockwise to point (x + dx. y + dy) that is close to (x, y) (Figure 6.49). The pivot also moves, from point (0, y) to nearby point (0, y + dy). How much does the wheel turn as a result of this motion? To answer this question, break the motion into two parts. First, roll the pivot along they-axis from (0. y) to (0, y + dy) without rotating the tracer arm. The tracer arm then ends up a point (x, y + dy) while maintaining a constant angle with the x-axis. Second, rotate the tracer arm by an angle d without moving the roller. Now the tracer is at point (x + dx, y + dy). Let 1 be the distance from the pivot to the wheel and let L be the distance from the pivot to the tracer (the length of the tracer arm). Figure 6.49 Mathematical analysis of the motion of the planimeter. 4. Show that x2+(yy)2=L2 Measuring Area from a Boundary: The Planimeter Figure 6.47 This magnetic resonance image of a patient’s brain shows a tumor, which is highlighted in red. (credit: modification of wood’ by Christaras A Wikimedia Commons) Imagine you are a doctor who has just received a magnetic resonance image of your patent’s brain. The brain has a tumor (Figure 6.47). How large is the tumor? To be precise, what is the area of the red region? The red cross-section of the tumor has an irregular shape, and therefore it is unlikely that you would be able o find a set of equations or inequalities for the region and then be able to calculate its area by conventional means. You could approximate the area by chopping the region into tiny squares (a Riemann sum approach), but this method always gives an answer with some error. Instead of trying to measure the area of the region directly, we can use a device called a rolling planimeter to calculate the area of the region exactly, simply by measuring its boundary. In this project you investigate how a planimeter works, and you use Green’s theorem to show the device calculates area correctly. A rolling planimeter is a device that measures the area of a planar region by tracing out the boundary of that region (Figure 6.48). To measure the area of a region, we simply run the tracer of the planimeter around the boundary of the region. The planimeter measures the number of turns through which the wheel rotates as we (race the boundary; the area of the shape is proportional to this number of wheel ruins. We can derive the precise proportionality equation using Green’s theorem. As the (racer moves around the boundary of the region, the tracer arm rotates and the roller moves back and forth (but does no; rotate). Figure 6.48 (a) A rolling planimeter. The pivot allows the tracer arm to rotate. The roller itself does not rotate; it only moves back and forth. (b) An interior view of a rolling planimeter. Notice that the wheel cannot rum if the planimeter is moving back and forth th the tracer arm perpendicular to the roller. Let C denote the boundary of region D. the area to be calculated. As the tracer traverses curve C, assume the roller moves along the y-axis (since the roller does not rotate, one can assume it moves along a straight line). Use the coordinates (x, y) to represent points on boundary C, and coordinates (0, y) to represent the position of the pivot. As the planimeter traces C. the pivot moves along the y-axis while the tracer arm rotates on the pivot. Watch a short animation (http://www.openstaxcollege.org/I/20_pianimeter) of a planimeter in action. Begin the analysis by considering the motion of the tracer as it moves from point (x, y) counterclockwise to point (x + dx. y + dy) that is close to (x, y) (Figure 6.49). The pivot also moves, from point (0, y) to nearby point (0, y + dy). How much does the wheel turn as a result of this motion? To answer this question, break the motion into two parts. First, roll the pivot along they-axis from (0. y) to (0, y + dy) without rotating the tracer arm. The tracer arm then ends up a point (x, y + dy) while maintaining a constant angle with the x-axis. Second, rotate the tracer arm by an angle d without moving the roller. Now the tracer is at point (x + dx, y + dy). Let 1 be the distance from the pivot to the wheel and let L be the distance from the pivot to the tracer (the length of the tracer arm). Figure 6.49 Mathematical analysis of the motion of the planimeter. 5. Assume the orientation of the planimeter is as shown in Figure 6.49. Explain why Yy. and use this inequality to show there is a unique value of y for each point Y=y=L2x.Measuring Area from a Boundary: The Planimeter Figure 6.47 This magnetic resonance image of a patient’s brain shows a tumor, which is highlighted in red. (credit: modification of wood’ by Christaras A Wikimedia Commons) Imagine you are a doctor who has just received a magnetic resonance image of your patent’s brain. The brain has a tumor (Figure 6.47). How large is the tumor? To be precise, what is the area of the red region? The red cross-section of the tumor has an irregular shape, and therefore it is unlikely that you would be able o find a set of equations or inequalities for the region and then be able to calculate its area by conventional means. You could approximate the area by chopping the region into tiny squares (a Riemann sum approach), but this method always gives an answer with some error. Instead of trying to measure the area of the region directly, we can use a device called a rolling planimeter to calculate the area of the region exactly, simply by measuring its boundary. In this project you investigate how a planimeter works, and you use Green’s theorem to show the device calculates area correctly. A rolling planimeter is a device that measures the area of a planar region by tracing out the boundary of that region (Figure 6.48). To measure the area of a region, we simply run the tracer of the planimeter around the boundary of the region. The planimeter measures the number of turns through which the wheel rotates as we (race the boundary; the area of the shape is proportional to this number of wheel ruins. We can derive the precise proportionality equation using Green’s theorem. As the (racer moves around the boundary of the region, the tracer arm rotates and the roller moves back and forth (but does no; rotate). Figure 6.48 (a) A rolling planimeter. The pivot allows the tracer arm to rotate. The roller itself does not rotate; it only moves back and forth. (b) An interior view of a rolling planimeter. Notice that the wheel cannot rum if the planimeter is moving back and forth th the tracer arm perpendicular to the roller. Let C denote the boundary of region D. the area to be calculated. As the tracer traverses curve C, assume the roller moves along the y-axis (since the roller does not rotate, one can assume it moves along a straight line). Use the coordinates (x, y) to represent points on boundary C, and coordinates (0, y) to represent the position of the pivot. As the planimeter traces C. the pivot moves along the y-axis while the tracer arm rotates on the pivot. Watch a short animation (http://www.openstaxcollege.org/I/20_pianimeter) of a planimeter in action. Begin the analysis by considering the motion of the tracer as it moves from point (x, y) counterclockwise to point (x + dx. y + dy) that is close to (x, y) (Figure 6.49). The pivot also moves, from point (0, y) to nearby point (0, y + dy). How much does the wheel turn as a result of this motion? To answer this question, break the motion into two parts. First, roll the pivot along they-axis from (0. y) to (0, y + dy) without rotating the tracer arm. The tracer arm then ends up a point (x, y + dy) while maintaining a constant angle with the x-axis. Second, rotate the tracer arm by an angle d without moving the roller. Now the tracer is at point (x + dx, y + dy). Let 1 be the distance from the pivot to the wheel and let L be the distance from the pivot to the tracer (the length of the tracer arm). Figure 6.49 Mathematical analysis of the motion of the planimeter. 6. Use step 5 to show that dY=dy+xL2x2dx.. cabmLwthaeaofapnregknD.useaplaiimewrtouacedaeboundayoftheregloameseaofthe region Is the length of the uacer am multiplied by the distance the wheel rolled.Measuring Area from a Boundary: The Planimeter Figure 6.47 This magnetic resonance image of a patient’s brain shows a tumor, which is highlighted in red. (credit: modification of wood’ by Christaras A Wikimedia Commons) Imagine you are a doctor who has just received a magnetic resonance image of your patent’s brain. The brain has a tumor (Figure 6.47). How large is the tumor? To be precise, what is the area of the red region? The red cross-section of the tumor has an irregular shape, and therefore it is unlikely that you would be able o find a set of equations or inequalities for the region and then be able to calculate its area by conventional means. You could approximate the area by chopping the region into tiny squares (a Riemann sum approach), but this method always gives an answer with some error. Instead of trying to measure the area of the region directly, we can use a device called a rolling planimeter to calculate the area of the region exactly, simply by measuring its boundary. In this project you investigate how a planimeter works, and you use Green’s theorem to show the device calculates area correctly. A rolling planimeter is a device that measures the area of a planar region by tracing out the boundary of that region (Figure 6.48). To measure the area of a region, we simply run the tracer of the planimeter around the boundary of the region. The planimeter measures the number of turns through which the wheel rotates as we (race the boundary; the area of the shape is proportional to this number of wheel ruins. We can derive the precise proportionality equation using Green’s theorem. As the (racer moves around the boundary of the region, the tracer arm rotates and the roller moves back and forth (but does no; rotate). Figure 6.48 (a) A rolling planimeter. The pivot allows the tracer arm to rotate. The roller itself does not rotate; it only moves back and forth. (b) An interior view of a rolling planimeter. Notice that the wheel cannot rum if the planimeter is moving back and forth th the tracer arm perpendicular to the roller. Let C denote the boundary of region D. the area to be calculated. As the tracer traverses curve C, assume the roller moves along the y-axis (since the roller does not rotate, one can assume it moves along a straight line). Use the coordinates (x, y) to represent points on boundary C, and coordinates (0, y) to represent the position of the pivot. As the planimeter traces C. the pivot moves along the y-axis while the tracer arm rotates on the pivot. Watch a short animation (http://www.openstaxcollege.org/I/20_pianimeter) of a planimeter in action. Begin the analysis by considering the motion of the tracer as it moves from point (x, y) counterclockwise to point (x + dx. y + dy) that is close to (x, y) (Figure 6.49). The pivot also moves, from point (0, y) to nearby point (0, y + dy). How much does the wheel turn as a result of this motion? To answer this question, break the motion into two parts. First, roll the pivot along they-axis from (0. y) to (0, y + dy) without rotating the tracer arm. The tracer arm then ends up a point (x, y + dy) while maintaining a constant angle with the x-axis. Second, rotate the tracer arm by an angle d without moving the roller. Now the tracer is at point (x + dx, y + dy). Let 1 be the distance from the pivot to the wheel and let L be the distance from the pivot to the tracer (the length of the tracer arm). Figure 6.49 Mathematical analysis of the motion of the planimeter. 7. Use Green’s theorem to show that cx L 2 x 2 dx=0.Measuring Area from a Boundary: The Planimeter Figure 6.47 This magnetic resonance image of a patient’s brain shows a tumor, which is highlighted in red. (credit: modification of wood’ by Christaras A Wikimedia Commons) Imagine you are a doctor who has just received a magnetic resonance image of your patent’s brain. The brain has a tumor (Figure 6.47). How large is the tumor? To be precise, what is the area of the red region? The red cross-section of the tumor has an irregular shape, and therefore it is unlikely that you would be able o find a set of equations or inequalities for the region and then be able to calculate its area by conventional means. You could approximate the area by chopping the region into tiny squares (a Riemann sum approach), but this method always gives an answer with some error. Instead of trying to measure the area of the region directly, we can use a device called a rolling planimeter to calculate the area of the region exactly, simply by measuring its boundary. In this project you investigate how a planimeter works, and you use Green’s theorem to show the device calculates area correctly. A rolling planimeter is a device that measures the area of a planar region by tracing out the boundary of that region (Figure 6.48). To measure the area of a region, we simply run the tracer of the planimeter around the boundary of the region. The planimeter measures the number of turns through which the wheel rotates as we (race the boundary; the area of the shape is proportional to this number of wheel ruins. We can derive the precise proportionality equation using Green’s theorem. As the (racer moves around the boundary of the region, the tracer arm rotates and the roller moves back and forth (but does no; rotate). Figure 6.48 (a) A rolling planimeter. The pivot allows the tracer arm to rotate. The roller itself does not rotate; it only moves back and forth. (b) An interior view of a rolling planimeter. Notice that the wheel cannot rum if the planimeter is moving back and forth th the tracer arm perpendicular to the roller. Let C denote the boundary of region D. the area to be calculated. As the tracer traverses curve C, assume the roller moves along the y-axis (since the roller does not rotate, one can assume it moves along a straight line). Use the coordinates (x, y) to represent points on boundary C, and coordinates (0, y) to represent the position of the pivot. As the planimeter traces C. the pivot moves along the y-axis while the tracer arm rotates on the pivot. Watch a short animation (http://www.openstaxcollege.org/I/20_pianimeter) of a planimeter in action. Begin the analysis by considering the motion of the tracer as it moves from point (x, y) counterclockwise to point (x + dx. y + dy) that is close to (x, y) (Figure 6.49). The pivot also moves, from point (0, y) to nearby point (0, y + dy). How much does the wheel turn as a result of this motion? To answer this question, break the motion into two parts. First, roll the pivot along they-axis from (0. y) to (0, y + dy) without rotating the tracer arm. The tracer arm then ends up a point (x, y + dy) while maintaining a constant angle with the x-axis. Second, rotate the tracer arm by an angle d without moving the roller. Now the tracer is at point (x + dx, y + dy). Let 1 be the distance from the pivot to the wheel and let L be the distance from the pivot to the tracer (the length of the tracer arm). Figure 6.49 Mathematical analysis of the motion of the planimeter. 8. Use step 7 t0 show that the total wheels roll is Total wheel roll = 1Lcxdy . it look a bit of work, him this equation says that the variable of Integral on Yin sap 3can be replaced with y.Measuring Area from a Boundary: The Planimeter Figure 6.47 This magnetic resonance image of a patient’s brain shows a tumor, which is highlighted in red. (credit: modification of wood’ by Christaras A Wikimedia Commons) Imagine you are a doctor who has just received a magnetic resonance image of your patent’s brain. The brain has a tumor (Figure 6.47). How large is the tumor? To be precise, what is the area of the red region? The red cross-section of the tumor has an irregular shape, and therefore it is unlikely that you would be able o find a set of equations or inequalities for the region and then be able to calculate its area by conventional means. You could approximate the area by chopping the region into tiny squares (a Riemann sum approach), but this method always gives an answer with some error. Instead of trying to measure the area of the region directly, we can use a device called a rolling planimeter to calculate the area of the region exactly, simply by measuring its boundary. In this project you investigate how a planimeter works, and you use Green’s theorem to show the device calculates area correctly. A rolling planimeter is a device that measures the area of a planar region by tracing out the boundary of that region (Figure 6.48). To measure the area of a region, we simply run the tracer of the planimeter around the boundary of the region. The planimeter measures the number of turns through which the wheel rotates as we (race the boundary; the area of the shape is proportional to this number of wheel ruins. We can derive the precise proportionality equation using Green’s theorem. As the (racer moves around the boundary of the region, the tracer arm rotates and the roller moves back and forth (but does no; rotate). Figure 6.48 (a) A rolling planimeter. The pivot allows the tracer arm to rotate. The roller itself does not rotate; it only moves back and forth. (b) An interior view of a rolling planimeter. Notice that the wheel cannot rum if the planimeter is moving back and forth th the tracer arm perpendicular to the roller. Let C denote the boundary of region D. the area to be calculated. As the tracer traverses curve C, assume the roller moves along the y-axis (since the roller does not rotate, one can assume it moves along a straight line). Use the coordinates (x, y) to represent points on boundary C, and coordinates (0, y) to represent the position of the pivot. As the planimeter traces C. the pivot moves along the y-axis while the tracer arm rotates on the pivot. Watch a short animation (http://www.openstaxcollege.org/I/20_pianimeter) of a planimeter in action. Begin the analysis by considering the motion of the tracer as it moves from point (x, y) counterclockwise to point (x + dx. y + dy) that is close to (x, y) (Figure 6.49). The pivot also moves, from point (0, y) to nearby point (0, y + dy). How much does the wheel turn as a result of this motion? To answer this question, break the motion into two parts. First, roll the pivot along they-axis from (0. y) to (0, y + dy) without rotating the tracer arm. The tracer arm then ends up a point (x, y + dy) while maintaining a constant angle with the x-axis. Second, rotate the tracer arm by an angle d without moving the roller. Now the tracer is at point (x + dx, y + dy). Let 1 be the distance from the pivot to the wheel and let L be the distance from the pivot to the tracer (the length of the tracer arm). Figure 6.49 Mathematical analysis of the motion of the planimeter. 9. Use Green’s theorem to that the area of D is cxdy. The logic is similar to the logic used to show that the area of DD=12cydx+xdx.]Measuring Area from a Boundary: The Planimeter Figure 6.47 This magnetic resonance image of a patient’s brain shows a tumor, which is highlighted in red. (credit: modification of wood’ by Christaras A Wikimedia Commons) Imagine you are a doctor who has just received a magnetic resonance image of your patent’s brain. The brain has a tumor (Figure 6.47). How large is the tumor? To be precise, what is the area of the red region? The red cross-section of the tumor has an irregular shape, and therefore it is unlikely that you would be able o find a set of equations or inequalities for the region and then be able to calculate its area by conventional means. You could approximate the area by chopping the region into tiny squares (a Riemann sum approach), but this method always gives an answer with some error. Instead of trying to measure the area of the region directly, we can use a device called a rolling planimeter to calculate the area of the region exactly, simply by measuring its boundary. In this project you investigate how a planimeter works, and you use Green’s theorem to show the device calculates area correctly. A rolling planimeter is a device that measures the area of a planar region by tracing out the boundary of that region (Figure 6.48). To measure the area of a region, we simply run the tracer of the planimeter around the boundary of the region. The planimeter measures the number of turns through which the wheel rotates as we (race the boundary; the area of the shape is proportional to this number of wheel ruins. We can derive the precise proportionality equation using Green’s theorem. As the (racer moves around the boundary of the region, the tracer arm rotates and the roller moves back and forth (but does no; rotate). Figure 6.48 (a) A rolling planimeter. The pivot allows the tracer arm to rotate. The roller itself does not rotate; it only moves back and forth. (b) An interior view of a rolling planimeter. Notice that the wheel cannot rum if the planimeter is moving back and forth th the tracer arm perpendicular to the roller. Let C denote the boundary of region D. the area to be calculated. As the tracer traverses curve C, assume the roller moves along the y-axis (since the roller does not rotate, one can assume it moves along a straight line). Use the coordinates (x, y) to represent points on boundary C, and coordinates (0, y) to represent the position of the pivot. As the planimeter traces C. the pivot moves along the y-axis while the tracer arm rotates on the pivot. Watch a short animation (http://www.openstaxcollege.org/I/20_pianimeter) of a planimeter in action. Begin the analysis by considering the motion of the tracer as it moves from point (x, y) counterclockwise to point (x + dx. y + dy) that is close to (x, y) (Figure 6.49). The pivot also moves, from point (0, y) to nearby point (0, y + dy). How much does the wheel turn as a result of this motion? To answer this question, break the motion into two parts. First, roll the pivot along they-axis from (0. y) to (0, y + dy) without rotating the tracer arm. The tracer arm then ends up a point (x, y + dy) while maintaining a constant angle with the x-axis. Second, rotate the tracer arm by an angle d without moving the roller. Now the tracer is at point (x + dx, y + dy). Let 1 be the distance from the pivot to the wheel and let L be the distance from the pivot to the tracer (the length of the tracer arm). Figure 6.49 Mathematical analysis of the motion of the planimeter. 10. Conclude that the sea of D equals the length of the air am multiplied by the total rolling distance of the wheel. You now know how a planimeter works and you have used Green’s theorem to justify that it works. To calculate the area of planar region D, use a planimeter to trace the boundary of the region .The area of region is the length of the tracer arm multiplied by the distance wheel rolled .For the following exercises, evaluate the line integrals by applying Green’s theorem. 146. c2xydx+(x+y)dy. Where C is the path from (0, 0) to (1, 1) along the graph of y = x3and from (1, 1) to (0, 0) along the graph of y = x oriented in the counterclockwise direction For the following exercises, evaluate the line integrals by applying Green’s theorem. 147. c2xydx+(x+y)dy. where C is the boundary of the region lying between the graphs of y = 0 and y = 4- x2oriented in the counterclockwise direction For the following exercises, evaluate the line integrals by applying Green’s theorem. 148. c2arctan( y x)dx+In( x 2+ y 2)dy,where C is defined by x=4+2cos,y=sinoriented in the counterclockwise direction For the following exercises, evaluate the line integrals by applying Green’s theorem. 149. csinxcosydx+(xy+cosxsiny)dy,where C is the boundary of the region lying between the graphs of y = x and y = x oriented in the counterclockwise direction For the following exercises, evaluate the line integrals by applying Green’s theorem. 150. c(ydx+xdy),where C consists of line segment C1from (- 1, 0) to (1, 0), followed by the semicircular arc C, from (1,0) back to (1, 0)For the following exercises, evaluate the line integrals by applying Green’s theorem. 150. cxydx+(x+y)dy,where C is the boundary of the region lying between the graphs of x2+ y2= 1 and x2+ y2= 9 oriented in the counterclockwise direction For the following exercises, use Green’s theorem. 152. Let C be the curve consisting of line segments from (0,0) to (1, 1) to (0, 1) and back to (0, 0). Find the value of cxydx+ y 2+1dy.For the following exercises, use Green’s theorem. 153. Evaluate line integral cxe2xdx+( x 4+2 x 2 y 2)dy,. where C is the boundary C of the region between circles x2+ y2= 1 and x2+ y2= 4, and is a positively oriented curve.Find the counterclockwise circulation of field F(x,y)=xyi+y2j around and over the boundary of the region enclosed by curves y = x2and y = x in the first quadrant and oriented in the counterclockwise direction.Evaluate cy3dxx3y2dy,where C is the positively oriented circle of radius 2 centered at the origin.Evaluate cy3dxx3dy,where C includes the two circles of radius 2 and radius 1 centered at the origin, both with positive orientation.Calculate cx2ydx+xy2dy,where C isa circle of radius 2 centered at the oligin and oriented in the counterclockwise direction.Calculate integral c2[y+xsin(x])dx+[x2cos(y)3y2]dyalong triangle C with vertices (0, 0), (1, 0) and (1, 1), oriented counterclockwise, using Green’s theorem.Evaluate integral c( x 2+ y 2)dx+2xydy,where C is the curve that follows parabola y=x2 from (0, 0)(2, 4). then the line from (2, 4) to (2,0), and finally the line from (2, 0) to (0,0).Evaluate line integralc(ysin( y)cos( y)dx+2x sin 2( y)dy,)where C is oriented in a counterclockwise path around the region bounded by x=1,x=2,y=4x2,and,y=x2 For the following exercises, use Green’s theorem to find the area. 161. Find the area between ellipse x29+y24=1 and circle x2+y2=25 For the following exercises, use Green’s theorem to find the area. 162. Find the area of the region enclosed by parametric equation p()=(cos()cos2())i+(sin()cos())jfor02.For the following exercises, use Green’s theorem to find the area. 163. Find the area of the region bounded by hypocycloid r(t)=cos3(t)i+sin3(t)j . The curve is parameterized by t[0,2].For the following exercises, use Green’s theorem to find the area. 164. Find the area of a pentagon with vertices (0, 4), (4, 1), (3, 0), (-1, -1), and (-2, 2).For the following exercises, use Green’s theorem to find the area. 165. Use Green’s theorem to evaluate c+( y 2+ x 3)dx+x4dy,where C+ is the perimeter of square [0, 1] × [0, 1] oriented counterclockwise.

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