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Bartleby Sitemap - Textbook Solutions

All Textbook Solutions for Calculus Volume 3

In the following exercises, calculate the integrals by interchanging the order of integration. 19. 46(21 3 x+ydy)dx In the following exercises, calculate the integrals by interchanging the order of integration. 20. 19(42 x y2 dy)dx In the following exercises, evaluate the iterated integrals by choosing the order of integration. 21. 00/2sin(2x)cos(3y)dxdy In the following exercises, evaluate the iterated integrals by choosing the order of integration. 22. /12/8/4/3[cotx+tan( 2y)]dxdy In the following exercises, evaluate the iterated integrals by choosing the order of integration. 23. 1e1e[1xsin( lnx)+1ycos( lny)]dxdy In the following exercises, evaluate the iterated integrals by choosing the order of integration. 24. 1e1esin( lnx)cos( lny)xydxdy In the following exercises, evaluate the iterated integrals by choosing the order of integration. 25. 1212( lny x+ x 2y+1)dydx In the following exercises, evaluate the iterated integrals by choosing the order of integration. 26. 1e12x2ln(x)dydx In the following exercises, evaluate the iterated integrals by choosing the order of integration. 27. 1312yarctan( 1 x)dydx In the following exercises, evaluate the iterated integrals by choosing the order of integration. 28. 0101/2(arcinx+arciny)dydx In the following exercises, evaluate the iterated integrals by choosing the order of integration. 29. 0112xex+4ydydx In the following exercises, evaluate the iterated integrals by choosing the order of integration. 30. 1201xexydydx In the following exercises, evaluate the iterated integrals by choosing the order of imtegration. 31. 1e1e( xlny y + lnx x )dydx In the following exercises, evaluate the iterated integrals by choosing the order of imtegration. 32. 1e1e( xlny y + ylnx x )dydx In the following exercises, evaluate the iterated integrals by choosing the order of integration. 33. 0112( x x2 +y2 )dydx In the following exercises, evaluate the iterated integrals by choosing the order of integration. 34. 0112yx+ y 2dydx function over the given rectangles. 35. f(x,y)=x+2y,R=[0,1][0,1]function over the given rectangles. 36. f(x,y)=x4+2y3,R=[1,2][2,3]function over the given rectangles. 37. f(x,y)=sinhx+sinhy,R=[0,1][2,3]function over the given rectangles. 38. f(x,y)=arctan(xy),R=[0,1][0,1]Let f and g be two continuous functions such that 0m1f(x)M1for any x[a,b] and 0m2g(y)M2, for any y[a,b]. Show that the following inequality is true:In the following exercises, use property y. of double integrals and the answer from the preceding exercise to show that the following inequalities are true. 40. 1e2RsinxcosydA248 where R=[0,1][0,1]In the following exercises, use property y. of double integrals and the answer from the preceding exercise to show that the following inequalities are true. 41. 2144RsinxcosydA248 where R=[0,2][0,2]In the following exercises, use property y. of double integrals and the answer from the preceding exercise to show that the following inequalities are true. 42. 2144ReycosxdA2 where R=[0,2][0,2]In the following exercises, use property y. of double integrals and the answer from the preceding exercise to show that the following inequalities are true. 43. 0R(lnx)(lny)dA( e1)2 where R=[1,e][1,e]Let f and g be two continuous functions such that in0m1f(x)M1for any x[a,b] and 0m2g(y)M2, for any y[c,d]. Show that the following inequality is true: (m1+m2)(ba)(cd)abcb[f(x)+g(y)]dydx( M 1+ M 2)(ba)(cd).In the following exercises, use property y. of double integrals and the answer from the preceding exercise to show that the following inequalities are true. 45. 2eR(e x 2 +e y 2 )dA2 where R=[0,1][0,1]In the following exercises, use property y. of double integrals and the answer from the preceding exercise to show that the following inequalities are true. 46. 236R(sinx+cosy)dA2336 where R=[6,3][6,3]In the following exercises, use property y. of double integrals and the answer from the preceding exercise to show that the following inequalities are true. 47. 2e/2R(cosx+r y)dA where R=[0,2][0,2]In the following exercises, use property y. of double integrals and the answer from the preceding exercise to show that the following inequalities are true. 48. 1eR(e ylnx)dA2 . Where R=[0,1][0,1]In the following exercises, the function f is given in terms of double integrals. a. Determine the explicit form of the function f. b. Find the volume of the solid tinder the surface z = f(x. v) and above the region R. C. Find the average value of the function f on R. d. Use a computer algebra system (CAS) to plot z = f(x. y) and z = favein the same system of coordinates. 49. [T] f(x,y)=0y0x(xs+yt)dsdtwhere (x,y)R=[0,1][0,1]In the following exercises, the function f is given in terms of double integrals. a. Determine the explicit form of the function f. b. Find the volume of the solid tinder the surface z = f(x. v) and above the region R. C. Find the average value of the function f on R. d. Use a computer algebra system (CAS) to plot z = f(x. y) and z = favein the same system of coordinates. 50. [T]f(x,y)=0x0y[cos(s)+cos(t)]dtds, where (x,y)R=[0,3][0,3]In the following exercises, the function f is given in terms of double integrals. a. Determine the explicit form of the function f. b. Find the volume of the solid tinder the surface z = f(x. v) and above the region R. C. Find the average value of the function f on R. d. Use a computer algebra system (CAS) to plot z = f(x. y) and z = favein the same system of coordinates. 51. Show that if f and g are continuous on [a,b] and [c,d], respectively, then cbad[ f( x )+g( y )]dydx=( dc) a b f(x )dx+abcdg(y)dydx=(ba)cdg( y)dy+cd c b f( x )dxdy.In the following exercises, the function f is given in terms of double integrals. a. Determine the explicit form of the function f. b. Find the volume of the solid tinder the surface z = f(x. v) and above the region R. C. Find the average value of the function f on R. d. Use a computer algebra system (CAS) to plot z = f(x. y) and z = favein the same system of coordinates. 52. Show that abcdy(x)+g(y)dx=12( d 2 c 2)( 0 b f(x )dx)+12(b2a2(cdg( y)dy))[T] Consider the function f(x,y)=ex2y2where (x,y)R=[1,1][1,1] . a. Use the midpoint rule with m = n = 2. 4.... 10 to estimate the double integral I= Rex2ydA. Round your answers to the R nearest hundredths. b. For m = n = 2. find the average value of f over the region R. Round your answer to the nearest hundredths.[T] Consider the function f(x,y)=sin(x2)cos(y2) . where (x. y) (x,y)R=[1,1][1,1] . a. Use the midpoint rule with m= n = 2. 4,. ... 10 to estimate the double integral I=Rsin(x2)cos(y2)dA . Round your answers to the nearest hundredths. b. For m = n = 2. find the average value of f over the region R. Round your answer to the nearest hundredths. c. Use a CAS to graph in the same coordinate system the solid whose volume is given by Rsin(x2)cos(y2)dA and the plane z=fave In the following exercises, the functions fnare given, where n1 is a natural number. a. Find the volume of the solids S,, under the surfaces z = f(x. v) and above the region R. b. Determine the limit of the volumes of the solids S as n increases without bound. 55. f(x,y)=xn+yn+xy,(x,y)R=[0,1][0,1]In the following exercises, the functions fnare given, where n1 is a natural number. a. Find the volume of the solids S,, under the surfaces z = f(x. y) and above the region R. b. Deteimine the limit of the volumes of the solids S as n increases without bound. 55. f(x,y)=xn+ynxy,(x,y)R=[0,1][0,1]In the following exercises, the functions fnare given, where n1 is a natural number. a. Find the volume of the solids S,, under the surfaces z = f(x. y) and above the region R. b. Deteimine the limit of the volumes of the solids S as n increases without bound. 56. f(x,y)=1xn+1yn(x,y)R=[1,2][0.1]In the following exercises, the functions fnare given, where n1 is a natural number. a. Find the volume of the solids S,, under the surfaces z = f(x. y) and above the region R. b. Deteimine the limit of the volumes of the solids S as n increases without bound. 58. Use the midpoint rule with m=n to show that the average value of a function f on a rectangular region R=[a,b][x,d] is approximated by fave1n2j=1nf( 1 2( x i1 ++xi ), 1 2( y j1 +yj ))An isotherm map is a chart connecting points having the same temperature at a given time for a given period of time. Use the preceding exercise and apply the midpoint rule with m = n = 2 to find the average temperature over the region given in the following figure.In the following exercises, specify whether the region is of Type I or Type II. 60. The region D bounded by y=x3,y=x3+1,x=0 . and x=1 as given in the following figure.In the following exercises, specify whether the region is of Type I or Type II. 61.Find the average value of the function f(x. y) = 3xy on the region graphed in the previous exercise.In the following exercises, specify whether the region is of Type I or Type II. 62. Find the area of the region D given in the previous exercise.In the following exercises, specify whether the region is of Type I or Type II. 63. The region D bounded by y = sin x. y = 1 + sin x. x = 0. and x=2 as given in following figure.wIn the following exercises, specify whether the region is of Type I or Type II. 64. Find the average value of the funcion f(x,y)=cosx on the region graphed in the previous exercise.In the following exercises, specify whether the region is of Type I or Type II. 65. Find the average value of the function f(x,y)=cosx on region graphed in the previous exercise.In the following exercises, specify whether the region is of Type I or Type II. 66. The region D bounded by x=y21 and x=1y2 as given in the following figure In the following exercises, specify whether the region is of Type I or Type II. 67. Find the volume of the solid under the graph of the function f(x, y) = xy + 1 and above the region in the figure in the previous exercise.In the following exercises, specify whether the region is of Type I or Type II. 68. The region D bounded by y=0,x=10+y and x=10y as given in the following figure.In the following exercises, specify whether the region is of Type I or Type II. 69. Find the volume of the solid under the graph of the function f(x,y)=x+y and above the region in the figure from the previous exercise.In the following exercises, specify whether the region is of Type I or Type II. 70. The region D bounded by y=0 . x=y1, x=2 - as given in the following figure.In the following exercises, specify whether the region is of Type I or Type II. 71. The region D bounded by y=0 and y=x21 as given in the following figure.In the following exercises, specify whether the region is of Type I or Type II. 72. Let D be the region bounded by the curves of equations y=x,y=x. and y=2x2. Explain why D is neither of Type I nor II.In the following exercises, specify whether the region is of Type I or Type II. 73. Let D be the region bounded by the curves of equation y=cosx and y=4x2 and x-axis Explain why D is neither of Type I nor II.In the following exercises, evaluate the double integral Df(x,y)dAover the region D. 74. f(x,y)=2x+5y and D={(x,y)0x1,x3yx3+1}In the following exercises, evaluate the double integral Df(x,y)dAover the region D. 75. f(x,y)=1 and D={(x,y)0x2,sinxy1+sinx}In the following exercises, evaluate the double integral Df(x,y)dAover the region D. 76. f(x,y)=2 and D={(x,y)1y1,y21xarccosy}In the following exercises, evaluate the double integral Df(x,y)dAover the region D. 77. f(x,y)=2 and D={(x,y)1y1,y21x1y2}In the following exercises, evaluate the double integral Df(x,y)dAover the region D. 78. f(x,y)=siny and D is the triangular region with vertices(0,0), (0,3), and (3,0)In the following exercises, evaluate the double integral Df(x,y)dAover the region D. 79. f(x,y)=x+1 and D is the triangular region with vertices (0,0),(0,2), and (2,2)Evaluate the iterated integrals. 80. 012x3x(x+ y 2)dydx Evaluate the iterated integrals. 81. 012x2x+1(v+lnu)dvdu Evaluate the iterated integrals. 82. ee2lnu2(v+lnu)dvdu Evaluate the iterated integrals. 83. 12u21u(8uv)dvdu Evaluate the iterated integrals. 84. 01 1 y 2 1 y 2 (2x+4 x 3)dxdy Evaluate the iterated integrals. 85. 01/2 14 y 2 14 y 2 4dxdy Evaluate the iterated integrals. 86. Let D be the region bounded by y=1x2,y=4x2 , and the x-and y-axes. a. Show that DxdA=011x24x2xdydx+1x24x2xdydx+ by Dividing the region D into two regions of Type I. b. Evaluate the integral DxdA .Evaluate the iterated integrals. 87. Let D be the region bounded by y = 1. y = x. y = In x, and the x -axis. a. Show that dividing D into two regions of Type I. b. Evaluate the integral if y dA.yEvaluate the iterated integrals. 88. a. Show that 0by dividing the region D into two regions of Type I, where —x.’ 2_x2}. b. Evaluate the integral if ydA. 1)Evaluate the iterated integrals. 89. a. Show that 0by dividing the region D into two regions of Type I, where —x.’ 2_x2}. b. Evaluate the integral if ydA. 1)The region D bounded by x=0,y=x5+1 , and S y=3x2 is shown in the following figure. Find the area A(D) of the region D.The legion D bounded by y = cos x. y = 4 cos x. and x = ± 3 is shown in the following figure. Find the area A(D) of the region Find the area A(D) of the region D={(x,y)y1x2,y4x2,y0,x0} .Let D be the region bounded by y = 1, y = x. y = In x. and the x -axis. Find the area A(D) of the region D.Find the average value of the function f(x. y) = sin y on the triangular region with vertices (0, 0). (0, 3). and (3, 0).Find the average value of the function f(x. y) =-x + 1 on tile triangular region with vertices (0. 0). (0. 2). and (2. 2).In the following exercises, change the order of integration and evaluate the integral. 96. 1/2x+10x+1sinxdydx In the following exercises, change the order of integration and evaluate the integral. 97. 01x11xxdydx In the following exercises, change the order of integration and evaluate the integral. 98. 10 y+1 y+1y2dxdy In the following exercises, change the order of integration and evaluate the integral. 99. 1/21/2 y 2 +1 y 2 +1ydxdy The region D is shown in the following figure. Evaluate the double integral D(x2+y)dAby using the easier order of integration.The region D is given in the following figure. Evaluate the double integral D(x2y2)dAby using the easier order of integration.Find the volume of the solid under the surface z=2x+y2and above the region bounded by y=x5and y=x .Find the volume of the solid tinder the plane z=3x+yand above the region determined by y=x7 and y=x.Find the volume of the solid tinder the plane z=xy and above the region bounded by x = tan y, x = —tan y. and x = 1.Find the volume of the solid under the surface z = x3and above the plane region bounded by x=siny,x=siny , and x= 1.Let g be a positive, increasing, and differentiable function on the interval [a. b]. Show that the volume of the solid under the surface z = g’(x) and above the region bounded by y = 0. y = g(x). x = a. and x = b is given by 12(g2(b)g2(a)) .Let g be a positive, increasing, and differentiable function on the interval [a, b]. and let k be a positive real number. Show that the volume of the solid under the surface z = g’(x) and above the region bounded by y = g(x), y= g(x) + k, x = a. and x = b is given by k(g(b) — g(a)).Find the volume of the solid situated in the first octant and determined by the planes z= 2. z=0,x+y=1,x=0 . And y = 0.Find the volume of the solid situated in the first octant and bounded by the planes x + 2y = 1. x=0, y=0, z=4, and z=0.Find the volume of the solid bounded by the planes x+y=1, xy=1, . x=0,z=0 and z=10.Find the volume of the solid bounded by the planes x+y=1,xy=1,x+y=1,xy=1,z=1andz=0.Let S1 and S2 , be the solids situated in the first octant under the planes x + y + z = 1 and x + y + 2z = 1. respectively, and let S be the solid situated between S1. S2. x = 0. and y= 0. a. Find the volume of the solid S1. b. Find the volume of the solid S2. c. Find the volume of the solid S by subtracting the volumes of the solids S1and S2.Let S and 5, be the solids situated in the first octant under the planes 2x+2y+z=2 and x+y+z=1 , respectively, and let S be the solid situated between S1,S2,x=0 . and y= 0. a. Find the volume of the solid S1. b. Find the volume of the solid S2. c. Find the volume of the solid S by subtracting the volumes of the solids S1and S2.Let S1 and S2 be the solids situated in the first octant under the plane x + y + z = 2 and under the sphere>x + y + z = 4. respectively. If the volume of the solid S, is determine the volume of the solid S2 situated 43 between S1and S2by subtracting the volumes of these solids.Let S1 and S2 be the solids situated in the first octant under the plane x+y+z=2 and bounded by the cylinder x2+y2=4 . respectively. a. Find the volume of the solid S1. b. Find the volume of the solid 52. c. Find the volume of the solid S situated between S1and S2by subtracting the volumes of the solids S1and S2.[T] The following figure shows the region D bounded by the curves y = Sin x. x=0. and y=x4 Use a graphing calculator or CAS to find the x -coordinates of the intersection points of the curves and to determine the area of the region D. Round your answers to six decimal places.[T] The region D bounded by the curves y=cosx,x=0 and y=x3 is shown in the following figure. Use a graphing calculator or CAS to find the x—coordinates of the intersection points of the curves and to determine the area of the region D. Round your answers to six decimal places.Suppose that (X. Y) is the outcome of an experiment that must occur in a particular region S in the xy —plane. In this context, the region S is called the sample space of the experiment and X and Y are random variables. If D is a region included in S. then the probability of (X. Y) being in D is defined as P[(X,Y)D]=Dp(x,y)dxdy. where p(x. y) is the joint probability density of the experiment. Here, p(x,y) is a nonnegative function for which Dp(x,y)dxdy=1 . Assume that a point (X, Y) is chosen arbitrarily in the square [0,3][0,3] with the probability density p(x,y)={0otherwise.19(x,y)[0,3][0,3]} , Find the probability that the point (X. Y) is inside the unit square and interpret the result.Consider X and Y two random variables of probability densities p1(x) and p 2(x). respectively. The random variables X and Y are said to be independent if their joint density function is given by p(x,Y) = p1(x)p2(y). At a drive—thru restaurant, customers spend, on average, 3 minutes placing their orders and an additional 5 minutes paying for and picking up their meals. Assume that placing the order and paying for/picking up the meal are two independent events X and Y. If the waiting times are modeled by the exponential probability densities p1(x)={0otherwise13ex/3x0,andp2(y)={1 5ey/5y0. otherwise. respectively, the probability that a customer will spend less than 6 minutes in the drive—thru line is given by P[X+y6]=Dp(x,y)dxdy, where D={(x,y)x0,y0,x+y6} . Find and interpret the result.[T] The Reuleaux triangle consists of an equilateral triangle and three regions, each of them bounded by a side of the triangle and an arc of a circle of radius s centered at the opposite vertex of the triangle. Show that the area of the Reuleaux triangle in the following figure of side length s is s22(3) .[T] Show that the area of the lunes of Alhazen, the two blue lunes in the following figure, is the same as tile area of the right triangle ABC. The outer boundaiies of tile hines are semicircies of diameters AB and AC. respectively, and the inner boundaries are formed by tile ciicumciicle of the tiiangle ABC.In the following exercises, express the region D in polar coordinates. 122. D is the region of the disk of radius 2 centered at the origin that lies in the first quadrant.In the following exercises, express the region D in polar coordinates. 123. D is the region between the circles of radius 4 and radius 5 centered at the origin that lies in the second quadrant.In the following exercises, express the region D in polar coordinates. 124. D is the region bounded by the y -axis and x=1y2 .In the following exercises, express the region D in polar coordinates. 125. D is the region bounded by the x -axis and y=2x2 .In the following exercises, express the region D in polar coordinates. 126. D={(x,y)x2+y24x}In the following exercises, express the region D in polar coordinates. 127. D={(x,y)x2+y24y}In the following exercises, the graph of the polar rectangular region D is given. Express D in polar coordinates.In the following exercises, the graph of the polar rectangular region D is given. Express D in polar coordinates.In the following exercises, the graph of the polar rectangular region D is given. Express D in polar coordinates.In the following exercises, the graph of the polar rectangular region D is given. Express D in polar coordinates.In the following exercises, the graph of the polar rectangular region D is given. Express D in polar coordinates. 132. In the following graph, the region D is situated below y = x and is bounded by x = 1 q x = 5. and y=0.In the following exercises, the graph of the polar rectangular region D is given. Express D in polar coordinates. 133. In the following graph, the region D is bounded by y = x and y= x2.In the following exercises, evaluate the double integral Rf(x,y)dA over the polar rectangular region D. 134. f(x,y)=x2+y2,D={(r,)3rr5,02}In the following exercises, evaluate the double integral Rf(x,y)dA over the polar rectangular region D. 135. f(x,y)=x+y,D={(r,)3r5,02}In the following exercises, evaluate the double integral Rf(x,y)dA over the polar rectangular region D. 136. f(x,y)=x2+xy,D={(r,)[1r2,2]}In the following exercises, evaluate the double integral Rf(x,y)dA over the polar rectangular region D. 137. f(x,y)=x4+y4,D={(r,)[1r2,322]}In the following exercises, evaluate the double integral Rf(x,y)dA over the polar rectangular region D. 138. f(x,y)=3x2+y2, where D={(r,)3r4,2}.In the following exercises, evaluate the double integral Rf(x,y)dA over the polar rectangular region D. 139. f(x,y)=x4+2x2y2+y4 where D={(r,)3r4,3}In the following exercises, evaluate the double integral Rf(x,y)dA over the polar rectangular region D. 140. f(x,y)=sin(arctanyx), where D={(r,)r2,63}In the following exercises, evaluate the double integral Rf(x,y)dA over the polar rectangular region D. 141. f(x,y)=arctan(yx) , where D={(r,)2r3,43}In the following exercises, evaluate the double integral Rf(x,y)dA over the polar rectangular region D. 142. Dex2+y2[1+2arctan( y x)]dA,D={(r,)13}In the following exercises, evaluate the double integral Rf(x,y)dA over the polar rectangular region D. 143. Dex2+y2[1+2arctan( y x)]dA,D={(r,)12,43}In the following exercises, the integrals have been converted to polar coordinates. Verify that the identities aie true and choose the easiest way to evaluate the integrals, in rectangular or polar coordinates. 144. 120x( x 2+ y 2)dydx=0 4 0 2sec r3 drd In the following exercises, the integrals have been converted to polar coordinates. Verify that the identities are true and choose the easiest way to evaluate the integrals, in rectangular or polar coordinates. 145. 230xx x2 +y2 dydx=0 /4tansecrcosdrd In the following exercises, the integrals have been converted to polar coordinates. Verify that the identities are true and choose the easiest way to evaluate the integrals, in rectangular or polar coordinates. 146. 01x2x1 x2 +y2 dydx=0 /4tan 0 sec drd In the following exercises, the integrals have been converted to polar coordinates. Verify that the identities are true and choose the easiest way to evaluate the integrals, in rectangular or polar coordinates. 147. 01x2xy x2 +y2 dydx=0 /4tan 0 sec rsindrd In the following exercises, convert the integrals to polar coordinates and evaluate them. 148. 030 9 y 2 ( x 2+ y 2)dxdy In the following exercises, convert the integrals to polar coordinates and evaluate them. 149. 02 4 y 2 4 y 2 ( x 2+ y 2)dxdy In the following exercises, convert the integrals to polar coordinates and evaluate them. 150. 010 1 x 2 (x+y)dydx In the following exercises, convert the integrals to polar coordinates and evaluate them. 151. 04 16 x 2 16 x 2 sin( x 2+ y 2)dydx Evaluate the integral DffrdAwhere D is the region bounded by the polar axis and the upper half of the cardioid r = 1 + cos .Find the area of the region D bounded by the polar axis and the upper half of the cardioid r=1+cos.Evaluate the integral DrdA, where D is the region bounded by the part of the four-leaved rose r = sin 2 situated in the first quadrant (see the following figure).Find the total area of the region enclosed by the four-leaved rose r = sin 2(see the figure in the previous exercise).Find the area of the region D, which is the region bounded by y=4x2,x=3,x=2 and y = 0.Find the area of the region D. which is the region inside the disk x2+y24 and to the tight of the line x= 1.Determine the average value of the function f(x. y) = x2+y2over the region D bounded by the polar curve r = cos 2. where 44(see the following graph).Determine the average value of the function f(x,y)=x2+y2 over the region D bounded by the polar curve r = 3 sin 2, where 02following graph).Find the volume of the solid situated in the first octant and bounded by the paraboloid z=14x24y2 and the planes x = 0. y = 0, and z = 0.Find the volume of the solid bounded by the paraboloid z=29x29y2 and the plane z = I.a. Find the volume of the solid S1 bounded by the cylinder x2+y2=1 and the planes z = 0 and b. Find the volume of the solid S2 outside the double cone z2=x2+y2. inside the cylinder x2+y2=1 . and above the plane z = 0. c. Find the volume of the solid inside the cone A) A) A) z2+x2+y2 and below the plane z = 1 by subtracting the volumes of the solid S1 and S2.a. Find the volume of the solid S1 inside the unit sphere x2+y2+z2=1 and above the plane z=0. b. Find the volume of the solid S2 inside the double cone (z-1 ) = x2+ y2and above the plane z=0. c. Find the volume of the solid outside the double cone (z — 1)2 = x2+ y2and inside the sphere x2+y2+z2=1 For the following two exercises, consider a spherical ring. which is a sphere with a cylindrical hole cut so that the axis of the cylinder passes through the center of the sphere (see the following figure). 164. If the sphere has radius 4 and the cylinder has radius 2. find the volume of the spherical ring.For the following two exercises, consider a spherical ring. which is a sphere with a cylindrical hole cut so that the axis of the cylinder passes through the center of the sphere (see the following figure). 165. A cylindrical hole of diameter 6 cm is bored through a sphere of radius 5 cm such that the axis of the cylinder passes through the center of the sphere. Find the volume of the resulting spherical ring.Find the volume of the solid that lies tinder the double cone z=4x2+4y2 inside the cylinder x2+y2=x, and above the plane z = 0.Find the volume of the solid that lies under the paraboloid z=x2+y2 inside the cylinder x2+y2=x . and above the plane z= 0.Find the volume of the solid that lies under the plane x +y‘+ z = 10 and above the disk x2+y2=4x.Find the volume of the solid that lies under the plane 2x+y+2z=8 and above the unit disk x2+y2=1 .A radial function f is a function whose value at each point depends only on the distance between that point and the origin of the system of coordinates: that is, f(x,y)=g(r). where r=x2+y2. Show that if f is a continuous radial function, then Df(x,y)dA=(21)[G( R 2)G( R 1)]. where G’(r) = rg(r) and (x,y)D={(r,)R1rR2,02} , with 0R1R2and 0122.Use the information from the preceding exercise to calculate the integral Dff(x2+y2)3dA. where D is the unit disk.Let f(x,y)=F(r)rbe a continuous radial function defined on the annular region D={(r,)R1R2,02} , where r=x2+y2,0R1,R2, and F is a differentiable function. Show that Df(x,yDA)=2[F( R 2)F( R 1)].Apply the preceding exercise to calculate the integral Dffe x 2+ y 2x2+y2dxdy , where D is the annular between the circles of radii 1and 2 situated in the third quadrant.Let f be a continuous function that can be expressed in polar coordinates as a function of 0 only: that is, f(x,y)=h(). where (x,y)D={r,}R1rR2,2with 0R1R2, and 0122 . Show that Df(x,y)dA=12(R22R12)H(2)H(1). where H is an antiderivative of h.Apply the preceding exercise to calculate the integral Dffy2x2dA. where D={(r,)1r2,63} .Let f be a continuous function that can be expressed in polar coordinates as a function of only; that is, f(x,y)=g(r)h(). where (x,y)D={(r,)R1rR2,12}, With 0R1R2, and 0122. Show that Df(x,y)dA=[G( R 2)F( R 1)][H( 2)H( 1)]. where G and H are antiderivatives of g and h. respectively.Evaluate Dff arctan (yx)x2+y2dA. where D={(r,)2r3,43} .A spherical cap is the region of a sphere that lies above or below a given plane. a. Show that the volume of the spherical cap in the figure below is 16h(3a2+h2) b. A spherical segment is the solid defined by intersecting a sphere with two parallel planes. If the distance between the planes is Ii. show that the volume of the spherical segment in the figure below is 16h(3a2+3b2+h2)In statistics, the joint density for two independent, normally distributed events with a mean =0 and a standard distiibution is defined by p(x,y)=122e Consider (X, Y). the Cartesian coordinates of a ball in the resting position after it was released from a position on the z-axis toward the xv -plane. Assume that the coordinates of the ball are independently normally distributed with a mean p = 0 and a standard deviation of c (in feet). The probability that the ball will stop no more than a feet from the origin is given by P[X2+Y2a2]=Dp(x,y)dydx. where D is the disk of radius a centered at the origin. Show that p[X2+Y2a2]=1ea2/22.The double improper integral e( x2 +y 2/2 )dxdymay be defined as the limit value of the double integrals Dae( x 2 + y 2 /2)dAover disks Daof radii a centered at the origin, as a increases without bound; that is, e( x2 )+ y 2/2dydx=alimDae( x2 +y 2/2 )dAa. Use polar coordinates to show that e( x2 +y 2/2 )dydx=2b. Show that ex2/2dx=2by using the relation emsp;e( x2 )+ y 2/2dydx=alimDae( x2 +y 2/2 )dA In the following exercises, evaluate the triple integrals over the rectangular solid box B. 181. B(2x+3y2+4z2)dV, where B={(x,y,z)zx1,0y2,0z3}In the following exercises, evaluate the triple integrals over the rectangular solid box B. 182. B(xy+yz+xz)dV , where B={(x,y,z)1x,2,0y2,1z3}In the following exercises, evaluate the triple integrals over the rectangular solid box B. 183. B(xcosy+z)dV, where B={(x,y,z)01,0y,1z1}In the following exercises, evaluate the triple integrals over the rectangular solid box B. 184. B(zsinx+y2)dv, where B={(x,y,z)0x,0y11z2}In the following exercises, change the order of integration by integrating first with respect to . then x. then y. 185. 011123( x2 +lny+z)dxdydz In the following exercises, change the order of integration by integrating first with respect to . then x. then y. 186. 011103( ze2 +2y)dxdz In the following exercises, change the order of integration by integrating first with respect to . then x. then y. 187. 121304( x2 z+1y )dxdydz In the following exercises, change the order of integration by integrating first with respect to . then x. then y. 188. 122101 x+y zdxdydz Let F. G and H be continuous functions on [a,b] [c,d] and [e,f] , respectively, where a,b,c,d,e and f are real number such that ab,cd , and ef .Show that abcdefF( x)G( y)H( z)dzdydx=( a b F( x )dx)(cdG( y)dy)(efH(z)dz).Let F. G. and H be differential functions on [a,b],[c,d]and [e,f] , respectively, where a. b. c, d. e, and f are real numbers such that a abcdefF( x)G( y)H( z)dzdydx=[F(b)(a)][G(d)G(c)][H(f)H(e)] .In the following exercises, evaluate the triple integrals over the bounded legion E={(x,y,z)axb,h1(x)yh2(x),ezf}191. E={(x,y,z)ax1,0yx+1,1z2}In the following exercises, evaluate the triple integrals over the bounded legion E={(x,y,z)axb,h1(x)yh2(x),ezf} 192. E(sinx+siny)dV , Where E={(x,y,z)1xe,0ycosx,1z1}In the following exercises, evaluate the triple integrals over the bounded legion E={(x,y,z)axb,h1(x)yh2(x),ezf} 193. E(sinx+siny)dV, where E={(x,y,z)01x2,cosxycosx,1z1}In the following exercises, evaluate the triple integrals over the bounded legion E={(x,y,z)axb,h1(x)yh2(x),ezf} 194. E(xy+yz+xz)dV , where E={(x,y,z)0x1,x2y2x2,0z1}In the following exercises, evaluate the triple integrals over the indicated bounded legion E. 195. E(x+2yz)dV where E={(x,y,z)0x1,0yx,0z5xy}In the following exercises, evaluate the triple integrals over the indicated bounded legion E. 196. E(x3+y3+z3)dV where E={(x,y,z)0x,2,y2x,0z4xy}In the following exercises, evaluate the triple integrals over the indicated bounded legion E. 197. Eydv where E={(x,y,z)1x1,1x2y1x2,0z1x2y2}In the following exercises, evaluate the triple integrals over the indicated bounded legion E. 198. EydV , where E={(x,y,z)2x241x2y4x2,0z4x2y2}In the following exercises, evaluate the triple integrals over the bounded region E of the form E={(x,y,z)1y2xy21y1,z2}. 199. E(sinx+y)dV where E={(x,y,z)y2xy21,y1,1z2}In the following exercises, evaluate the triple integrals over the bounded region E of the form E={(x,y,z)1y2xy21y1,z2}. 200. E(sinx+y)dV where E={(x,y,z)y4xy,0y1x,1z1}In the following exercises, evaluate the triple integrals over the bounded region E of the form E={(x,y,z)1y2xy21y1,z2}. 201. E(xyz)dV, where E={(x,y,z)y6xy,0y1x,1z1}In the following exercises, evaluate the triple integrals over the bounded region E of the form E={(x,y,z)1y2xy21y1,z2}. 202. EzdV where E={(x,y,)22yx2+y,0y1x,2z3}In the following exercises, evaluate the triple integrals over the bounded region E={(x,y,z)g1(y)xg2(y),cyd,u1(x,y)zu2(x,y)}203. EzdV where E={(x,y,z)yxy,0y12x,0z1x2y2}In the following exercises, evaluate the triple integrals over the bounded region E={(x,y,z)g1(y)xg2(y),cyd,u1(x,y)zu2(x,y)} 204. E(xz+1)dV where E={(x,y,z)1y2xy,0y12x,0z1x2y2}In the following exercises, evaluate the triple integrals over the bounded region E={(x,y,z)g1(y)xg2(y),cyd,u1(x,y)zu2(x,y)} 205. E(xz+1)dV , Where E={(x,y,z)1y2xy,0y12x,0z1x2y2}In the following exercises, evaluate the triple integrals over the bounded region E={(x,y,z)g1(y)xg2(y),cyd,u1(x,y)zu2(x,y)}206. E(xz+1)dV. where E={(x,y,z)g1(y)xg2(y),cyd,u1(x,y)}In the following exercises, evaluate the triple integrals over the bounded region E={(x,y,z)(x,y)D,u1(x,y)xzu2(x,y)}, where D is the projection of E onto the xy —plane. 207. D( 1 2 ( x+z )dz)dA. where D={(x,y)x2+y21}In the following exercises, evaluate the triple integrals over the bounded region E={(x,y,z)(x,y)D,u1(x,y)xzu2(x,y)}, where D is the projection of E onto the xy —plane. 208. D( 1 3 x( z+1 )dz)dA, where D={(x,y)x2y21,x5}In the following exercises, evaluate the triple integrals over the bounded region E={(x,y,z)(x,y)D,u1(x,y)xzu2(x,y)}, where D is the projection of E onto the xy —plane. 209. D( 0 10xy ( x+2z )dz)dA, where D={(x,y)y,x0,x+y10}In the following exercises, evaluate the triple integrals over the bounded region E={(x,y,z)(x,y)D,u1(x,y)xzu2(x,y)}, where D is the projection of E onto the xy —plane. 210. D( 0 4x2 +4y2 ydz)dA where D={(x,y)x2+y24,y1,x0}The solid E bounded by y2+z2=9,x=0 . x = 5 is shown in the following figure. Evaluate the integral EzdVby integrating first with respect to z then y, and then x.The solid E bounded by y=x,x=4,y=0 , and z = 1 is given in the following figure. Evaluate the integral ExyzdV by integrating first with respect to x, then y, and then z.[T] The volume of a solid E is given by the integral 2f0xf00fx2+y2dxdydx. Use a computer algebra system (CAS) to graph E and find its volume. Round your answer to two decimal places.[T] The volume of a solid E is given by the integral 1f0x2f00f0+x2+y2dzdydx. Use a CAS to graph E and find its volume V. Round your answer to two decimal places.In the following exercises, use two circular permutations of the variables x. y. and z to write new integrals whose values equal the value of the original integral. A circular permutation of x. y. and z is the arrangement of the numbers in one of the following orders: y, z, and x or z, x. and y. 215. 011324( x2 z2 +1)dxdydz In the following exercises, use two circular permutations of the variables x. y. and z to write new integrals whose values equal the value of the original integral. A circular permutation of x. y. and z is the arrangement of the numbers in one of the following orders: y, z, and x or z, x. and y. 216. 13010 x+1( 2x+5y+7z)dydxdz In the following exercises, use two circular permutations of the variables x. y. and z to write new integrals whose values equal the value of the original integral. A circular permutation of x. y. and z is the arrangement of the numbers in one of the following orders: y, z, and x or z, x. and y. 217. 01yy1x40 y 4 lnxdzdxdy In the following exercises, use two circular permutations of the variables x. y. and z to write new integrals whose values equal the value of the original integral. A circular permutation of x. y. and z is the arrangement of the numbers in one of the following orders: y, z, and x or z, x. and y. 218. 1101 y 6 y ( x+yz)dxdydz Set up the integral that gives the volume of the solid E bounded by y2=x2+z2and y=a2, where a>0.Set up the integral that gives the volume of the solid E bounded by z=y2+z2and x=a2, where a0 .Find the average value of the function f(x. y, z) = x + y + z over the parallelepiped determined by x = 0. x = 1. v = 0. y = 3 z = 0. and z= 5.Find the average value of the function f(x,y,z)=xyz over the solid E=[0,1][0,1][0,1] situated in the first octant.Find the volume of the solid E that lies under the plane x + y + z = 9 and whose projection onto the xy -plane is bounded by x=y1,x=0 . and x + y = 7.Find the volume of the solid E that lies under the plane 2x + y+ z = 8 and whose projection onto the xy -plane is bounded by x = sin-1 y. y= 0. and x=2 .Consider the pyramid with the base in the xv -plane of [2,2][2,2] and the vertex at the point (0. 0. 8). a. Show that the equations of the planes of the lateral faces of the pyramid are 4y + z = 8. 4yz=8,4x+z=8 . and 4x+z=8 . b. Find the volume of the pyramid.Consider the pyramid with the base in the xy -plane of [3,3][3,3] and the vertex at the point (0. 0. 9). a. Show that the equations of the planes of the side faces of the pyramid are 3y + z = 9. 3y+z=9,y=0andx=0 b. Find the volume of the pyramid.The solid E bounded by the sphere of equation x2+y2+z2=r2 with r > 0 and located in the first octant is represented in the following figure. a. Write the triple integral that gives the volume of z by integrating first with respect to z then with y, and then with x. b. Rewrite the integral in part a. as an equivalent integral in five other orders.The solid E bounded by the equation 9x2+4y2+z2=1 and located the first octant is represented in the following figure. a. Trite the triple integral that gives the volume of E by integrating first with respect to z. then with v, and then with x. b. Rewrite the integral in part a. as an equivalent integral in five other orders.Find the volume of the prism with vertices (0, 0. 0), (2. 0. 0), (2. 3, 0), (0, 3, 0), (0. 0. 1), and (2, 0. 1).Find the volume of the prism with vertices (0. 0. 0). (4, 0, 0), (4, 6. 0). (0, 6, 0), (0, 0, 1), and (4. 0. 1).The solid E bounded by z= 10—2x—y and situated in the first octant is given in the following figure. Find the volume of the solid.The solid E bounded by z=1x2 and situated in the first octant is given in the following figure. Find the volume of the solid.The midpoint rule for the triple integral Bf(x,y,z)dVover the rectangular solid box B is a generalization of the midpoint rule for double integrals. The region B is divided into subboxes of equal sizes and the integral is approximated by the triple Riemann sum i=1lj=1m k=1nf( x i , x j , z k )Vis (xi,yj,zk) the center of the box and V is the volume of each subbox. Apply the midpoint rule to approximate Bx2dV over the solid B={(x,y,z)0x1,0y1,0z1} by using a partition of eight cubes of equal size. Round your answer to three decimal places.[T] a. Apply the midpoint rule to approximate Bex2dvover the solid B={(x,y,z)0x1,0y1,0z1} by using a partition of eight cubes of equal size. Round your answer to three decimal places. b. Use a CAS to improve the above integral approximation in the case of a partition of n3cubes of equal size, where n = . 3,4..... 10.Suppose that the temperature in degrees Celsius at a point (x. y. z) of a solid E bounded by the coordinate planes and x + y + z = 5 is T(x. y. z) = xz + 5z + 10. Find the average temperature over the solid.Suppose that the temperature in degrees Fahrenheit at a point (x, y, z) of a solid E bounded by the coordinate planes and x +y + z = 5 is T(x. y, z) = x + y + xy. Find the average temperature over the solid.Show that the volume of a right square pyramid of height h and side length a is v=ha23 by using triple integrals.Show that the volume of a regular right hexagonal prism of edge length a is 3a332 by using triple integrals.Show that the volume of a regular right hexagonal pyramid of edge length a is a332 by using triple integrals.If the charge density at an arbitraiy point (x, y. z) of a solid E is given by the function (x,y,z)then the total charge inside the solid is defined as the triple integral E(x,y,z)dV. Assume that the charge density of the solid E enclosed by the paraboloids x=5y2z2 and x=y2z25is equal to the distance from an aibitraiy point of E to the origin. Set tip the integral that gives the total charge inside the solid E.Hot air balloons Rot all ballooning is a relaxing. peaceful pastime that many people enjoy. Many balloonist gatherings take place around the world, Such as the Albuquerque International Balloon Fiesta. The Albuquerque event is the largest hot air balloon festival in the world, with ovet 500 balloons participating each year. Figure 5.61 Balloons lift off at the 2001 Albuquerque international Balloon Fiesta. (credit: David Herrera, Flickr) As the name implies, hot air balloons use hot air to generate lift. (Hot air is less dense than cooler air, so the balloon floats as long as the hot air stays hot.) The heat is generated by a propane burner suspended below the opening of the basket. Once the balloon takes off, the pilot controls the altitude of the balloon, either by using the burner to heat the air and ascend or by using a vent near the top of the balloon to release heated air and descend. The pilot has very little control over where the balloon goes, however—balloons are at the mercy of the winds. The uncertainty over where we will end up is one of the reasons balloonists are attracted to the sport. In this project we use tnple integrals to learn more about hot air balloons. We model the balloon in two pieces. The top of the balloon is modeled by a half sphere of radius 28 feet. The bottom of the balloon is modeled by a fnistum of a cone (think of an ice cream cone with the pointy end cut off). The radius of the large end of the fnistum is 28 feet and the radius of the small end of the fnistum is 6 feet. A graph of our balloon model and a cross-sectional diagram showing the dimensions are shown in the following figure. FIgure 5.62 (a)Useahalfsphetetomodeltheroppartoftheballoonandafrnsnunofaconeomodel the bonom pan of the balloon. (b) A cross section of the balloon shong its dimensions.Hot air balloons Rot all ballooning is a relaxing. peaceful pastime that many people enjoy. Many balloonist gatherings take place around the world, Such as the Albuquerque International Balloon Fiesta. The Albuquerque event is the largest hot air balloon festival in the world, with ovet 500 balloons participating each year. Figure 5.61 Balloons lift off at the 2001 Albuquerque international Balloon Fiesta. (credit: David Herrera, Flickr) As the name implies, hot air balloons use hot air to generate lift. (Hot air is less dense than cooler air, so the balloon floats as long as the hot air stays hot.) The heat is generated by a propane burner suspended below the opening of the basket. Once the balloon takes off, the pilot controls the altitude of the balloon, either by using the burner to heat the air and ascend or by using a vent near the top of the balloon to release heated air and descend. The pilot has very little control over where the balloon goes, however—balloons are at the mercy of the winds. The uncertainty over where we will end up is one of the reasons balloonists are attracted to the sport. In this project we use tnple integrals to learn more about hot air balloons. We model the balloon in two pieces. The top of the balloon is modeled by a half sphere of radius 28 feet. The bottom of the balloon is modeled by a fnistum of a cone (think of an ice cream cone with the pointy end cut off). The radius of the large end of the fnistum is 28 feet and the radius of the small end of the fnistum is 6 feet. A graph of our balloon model and a cross-sectional diagram showing the dimensions are shown in the following figure. FIgure 5.62 (a)Useahalfsphetetomodeltheroppartoftheballoonandafrnsnunofaconeomodel the bonom pan of the balloon. (b) A cross section of the balloon shong its dimensions.Hot air balloons Rot all ballooning is a relaxing. peaceful pastime that many people enjoy. Many balloonist gatherings take place around the world, Such as the Albuquerque International Balloon Fiesta. The Albuquerque event is the largest hot air balloon festival in the world, with ovet 500 balloons participating each year. Figure 5.61 Balloons lift off at the 2001 Albuquerque international Balloon Fiesta. (credit: David Herrera, Flickr) As the name implies, hot air balloons use hot air to generate lift. (Hot air is less dense than cooler air, so the balloon floats as long as the hot air stays hot.) The heat is generated by a propane burner suspended below the opening of the basket. Once the balloon takes off, the pilot controls the altitude of the balloon, either by using the burner to heat the air and ascend or by using a vent near the top of the balloon to release heated air and descend. The pilot has very little control over where the balloon goes, however—balloons are at the mercy of the winds. The uncertainty over where we will end up is one of the reasons balloonists are attracted to the sport. In this project we use tnple integrals to learn more about hot air balloons. We model the balloon in two pieces. The top of the balloon is modeled by a half sphere of radius 28 feet. The bottom of the balloon is modeled by a fnistum of a cone (think of an ice cream cone with the pointy end cut off). The radius of the large end of the fnistum is 28 feet and the radius of the small end of the fnistum is 6 feet. A graph of our balloon model and a cross-sectional diagram showing the dimensions are shown in the following figure. FIgure 5.62 (a)Useahalfsphetetomodeltheroppartoftheballoonandafrnsnunofaconeomodel the bonom pan of the balloon. (b) A cross section of the balloon shong its dimensions. 3. Find the average temperature of the air in the balloon after the pilot has actIvated the burner br tO seconds.In the following exercises, evaluate the triple integrals Ef(x,y,z)dVover the solid E. 241. f(x,y,z)=z,B{(x,y,z)x2+y29,x0,y0,0z1}In the following exercises, evaluate the triple integrals Ef(x,y,z)dVover the solid E. 242. f(x,y,z)=xz2,B={(x,y,z)x2+y216,x0,y0,1z1}In the following exercises, evaluate the triple integrals Ef(x,y,z)dVover the solid E. 243 f(x,y,z)=xy,B={(x,y)x2+y21,x0,xy,1z1}In the following exercises, evaluate the triple integrals Ef(x,y,z)dVover the solid E. 244. f(x,y,z)=x2+y2,B={(x,y,z)x2+y24,x0,xy,0z3}In the following exercises, evaluate the triple integrals Ef(x,y,z)dVover the solid E. 245. f(x,y,z)=ex2+y2,B={(x,y,z)1x2+y24,y0,xy3,2z3}In the following exercises, evaluate the triple integrals Ef(x,y,z)dVover the solid E. 246. f=(x,y,z)=x2+y2,B={(x,y,z)1x2+y29,y0,0z1}a. Let B be a cylindrical shell with inner radius a, outer radius b. and height c. where 0 < a < b and c> 0. Assume that a function F defined on B can be expressed in cylindrical coordinates as F(x. , z) = f(r) + Iz(z). where f and Ii are b differentiable functions. If ff(r)dr = 0 and Iz(0) = 0. where f and Ii are antiderivatives of f and Ii. respectively, show that I F(x. y, z)dV = 2xc(bf(b) — af(a)) + x(b2 — a2)h(c). b. Use the previous result to show that Ill (z + sinR ‘(hdvd = 6,r2(,r —2), where B is a cylindrical shell with inner radius n. outer radius 2jr, and height 2.a. Let B be a cylindrical shell with inner radius a. outer radius b. and height c, where 0 < a < b and c> 0. Assume that a function F defined on B can be expressed in cylindrical coordinates as F(x. v z) = f(r)g( )h(z). where f. g. and Ii are differentiable functions. If abf(r)dr=0 . where is an antiderivative of f. show that J1 F(x. y. z)dV = [b?(b) — af(a)]I(2x) — — 11(0)]. where and Ii are antiderivatives of g and Ii. respectively. b. Use the previous result to show that z sinLt + v2dx dv dz = —1 2,r2. where B B is a cylindrical shell ‘with inner radius n’, outer radius 2,r. and height 2.In the following exercises, the boundaries of the solid E are given in cylindrical coordinates. a. Express the region E in cylindrical coordinates. b. Convert the integral Ef(x,y,z)dVto cylindrical coordinates. 249. E is bounded by the right circular cylinder r = 4 sin , the r-plane, and the sphere r2+ z2= 16.In the following exercises, the boundaries of the solid E are given in cylindrical coordinates. a. Express the region E in cylindrical coordinates. b. Convert the integral Ef(x,y,z)dVto cylindrical coordinates. 250. E is bounded by the right circular cylinder r=cos , the r -plane, the sphere r2+z2=9 .In the following exercises, the boundaries of the solid E are given in cylindrical coordinates. a. Express the region E in cylindrical coordinates. b. Convert the integral Ef(x,y,z)dVto cylindrical coordinates. 251. E is located in the first octant and is bounded by the circular paraboloid z = 9 — 3r2. the cylinder r=3 and the plane r(cos+sin)=20z .In the following exercises, the boundaries of the solid E are given in cylindrical coordinates. a. Express the region E in cylindrical coordinates. b. Convert the integral Ef(x,y,z)dVto cylindrical coordinates. 252. E is located in the first octant outside the circular paraboloid z = 10 — 2r2and inside the cylinder r=5and is bounded also by the plane z = 20 and =4 .In the following exercises, the function f and region E aie given. a. Express the region E and the function f in cylindrical coordinates. b). Convert the integral Bf(x,y,z)dV into cylindrical coordinates and evaluate it. 253. f(x,y,z)=1x+3,E={(x,y,z)0x2+y29,x0,y0,0zx+3}In the following exercises, the function f and region E aie given. a. Express the region E and the function f in cylindrical coordinates. b). Convert the integral Bf(x,y,z)dV into cylindrical coordinates and evaluate it. 254. f(x,y,z)=x2+y2,E={(x,y,z)0x2+y24,y0,0z3x}In the following exercises, the function f and region E aie given. a. Express the region E and the function f in cylindrical coordinates. b). Convert the integral Bf(x,y,z)dV into cylindrical coordinates and evaluate it. 255. f(x,y,z)=x2+y2,E={(x,y,z)1y2+z29,0x1y2z2}In the following exercises, the function f and region E aie given. a. Express the region E and the function f in cylindrical coordinates. b). Convert the integral Bf(x,y,z)dV into cylindrical coordinates and evaluate it. 256 f(x,y,z)=y,E{(x,y,z)1x2+z29,0y1x2z2} .In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. 257. E is above the xv -plane, inside the cylinder x2+y2=1 and below the plane z = 1.In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. 258. E is below the plane z = 1 and inside the paraboloid z = x2+ y2.In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. 259. E is bounded by the circular cone z=x2+y2 and z=1.In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. 260. E is located above the xy -plane, below z= 1, outside the one-sheeted hyperboloid x2+y2-z2= 1. and inside the cylinder x2+ y2= 2.In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. 261. E is located inside the cylinder x2+ y2= 1 and between the circular paraboloids z = 1 — x2— y2and z = x2+ y2.In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. 262. E is located inside the sphere x2+ y2+ z2= 1. above the xy -plane, and inside the circular cone z=x2+y2 In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. 263. E is located outside the circular cone x2+ y2 = (z-1 )2 and between the planes z = 0 and z = 2.In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. 264. E is located outside the circular cone z=1x2+y2 , above the xy-plane, and between the planes z=0 and z=2 .[T] Use a computer algebra system (CAS) to graph the solid whose volume is given by the iterated integral in cylindrical coordinates /2/201rdzdrd. Find the volume V of the solid. Round your answer to four decimal places.[T] Use a CAS to graph the solid whose volume is given by the iterated integral in cylindrical coordinates 0f/20f1r4frrdzdrd. Find the volume V of the solid Round your answer to four decimal places.267. Convert the integral into an integral in cylindrical coordinates.Convert the integral 020x 01 ( xy+z) dzdxdy into an integral in cylindrical coordinates. In the following exercises, evaluate the triple integral Bffff(x,y,z)dv over the solid B.f(x,y,z)=1,B={(x,y,z)x2+y2+z290,z0}270. f(x,y,z)=1x2+y2+z2,B={(x,y,z)x2+y2+z29,y0,z0}f(x,y,z)=x2+y2. B is bounded above by the half-sphere x2+y2+z2=9 with z0 and below by the cone 2z2=x2+y2.f(x. y, z) = z. B is bounded above by the half sphere x2+ y2+ z2= 16 with z0 and below by the cone 2z2=x2+y2.Show that if F(,,)=f()g()h() is a continuous function on the spherical box B={(,)}[a,b,a,], then BdfV=( a b 2 f( )dr)(ag( )d)(h()sind)a. A function F is said to have spherical svmmetiy if it depends on the distance to the origin only, that is, it can be expressed in spherical coordinates as F(x. y. z) = f(p). where =x2+y2+z2 . Show that BF(x,y,z)dV=2ab2f()d,where B is the region between the upper concentric hemispheres of radii a and b centered at the origin, with 0 < a < b and F a spherical function defined on B. b. Use the previous result to show that B(x2+y2+z2)x2+y2+z2dV=21 where B={(x,y,z)1x2+y2+z22,z0}a. Let B be the region between the upper concentric hemispheres of radii a and b centered at the origin and situated in the first octant, where 0 < a < b. Consider F a function defined on B whose form in spherical coordinates ( ,, ) is F(x,y,z)=f()cos . Show that if g(a)=g(b)=0 and abh(h)d=0 . then BF(x,y,z)dV=24[ah(a)bh(b)]where g is an antiderivative of f and h is an antiderivative of g. b. Use the previous result to show that Bzcos x 2 + y 2 + z 2 x 2 + y 2 + z 2 dV=32. where B is the region between the upper concentric hemispheres of radii and 2 centered at the origin and situated in the first octant.In the following exercises, the function f and region E are given. a. Express the region E and function f cylindrical coordinates. b. Convert the integral Bf(x,y,z)dV into cylindrical coordinates and evaluate it. 276. f(x,y,z)=z;E={(x,y,z)0x2+y2+z21,z0}In the following exercises, the function f and region E are given. a. Express the region E and function f cylindrical coordinates. b. Convert the integral Bf(x,y,z)dV into cylindrical coordinates and evaluate it. 277. E={(x,y,z)1x2+y2+z22,z0,y0}In the following exercises, the function f and region E are given. a. Express the region E and function f cylindrical coordinates. b. Convert the integral Bf(x,y,z)dV into cylindrical coordinates and evaluate it. 278. f(x,y,z)=2xy;E{(x,y,z)x2+y2z1x2y2,x0,y0}In the following exercises, the function f and region E are given. a. Express the region E and function f cylindrical coordinates. b. Convert the integral Bf(x,y,z)dV into cylindrical coordinates and evaluate it. 279. E={(x,y,z)x2+y2+z22z0,x2+y2z}In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. 280. E={(x,y,z)x2+y2z16x2y2,x0,y0}In the following exercises, find the volume of the solid E whose boundaries are given in rectangular coordinates. 280. E={(x,y,z)x2+y2z16x2y2,x0,y}Use spherical coordinates to find the volume of the solid situated outside the sphere = 1 and inside the sphere = cos . with [0,2] .Use spherical coordinates to find the volume of the ball 3 that is situated between the cones =4 and =3 .Convert the integral f44f16 y 216y2f16 x 2 y 216x2y2(x2+y2+z2)dz,dx,dy into an integral in spherical coordinates.Convert the integral 2f24 x 2f4x2 x 2+ y 2f16x2y2dzdydx into an integral in spherical coordinates.Convert the integral 2f24 x 2f4x2 x 2+ y 2f16x2y2dzdydx into an integral in 1. spherical coordinates and evaluate it.[T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates /2/6/6022 sin dpdd. Find the volume V of the solid. Round your answer to three decimal places.[T] Use a CAS to graph the solid whose volume is >given by the iterated integral in spherical coordinates as /2/6/6022sin p dp thp do. Find the volume V of the solid. Round your answer to three decimal places.[T] Use a CAS to evaluate the integral Efff(x2+y2)dV where E lies above the paraboloid z=x2+y2and below the plane z=3y .[T] a. Evaluate the integral Ee x 2 + y 2 + z 2 dV, where E is bounded by the spheres 4x2+4y2+4z2=1 and x2+y2+z2=1 . b. Use a CAS to find an approximation of the previous integral. Round your answer to two decimal places.Express the volume of the solid inside the sphere x2+y2+z2=16 and outside the cylinder x2+y2=4 as triple integrals in cylindrical coordinates and spherical coordinates, respectively.Express the volume of the solid inside the sphere x2+y2+z2=16 and outside the cylinder x2+y2=4 that is located in the first octant as triple integrals in cylindrical coordinates and spherical coordinates, respectively.The power emitted by an antenna has a power density per unit volume given in spherical coordinates by p(,,)=P02cos2sin4 . where P0is a constant with units in watts. The total power within a sphere B of radius r meters is defined as P=Bp(,,)dV. Find the total power P.Use the preceding exercise to find the total power a sphere B of radius 5 meters when the power density per unit volume is given by p(,,)=302cos2sin4 .A charge cloud contained in a sphere B of radius r centimeters centered at the origin has its charge density given by (x,y,z)=kx2+y2+z2Ccm3where k>0. The total charge contained in B is given by Q=Bq(x,y,z)dV. Find the total charge Use the preceding exercise to find the total charge cloud contained in the unit sphere if the charge density is q=(x,y,z)=20x2+y2+z2Ccm3 .In the following exercises, the region R occupied by a lamina is shown in a graph. Find the mass of R with the density function . 297. R is the triangular region with vertices (0. 0). (0. 3). and (6. 0): (x. y) = xy.In the following exercises, the region R occupied by a lamina is shown in a graph. Find the mass of R with the density function . 298. R is the triangular region with vertices (0, 0), (1, 1), (0, 5); p(x. y)= x+y.In the following exercises, the region R occupied by a lamina is shown in a graph. Find the mass of R with the density function . 299. R is the rectangular (0. 0). (0, 3), (6. 3), and (6. 0); (x,y)=xy In the following exercises, the region R occupied by a lamina is shown in a graph. Find the mass of R with the density function . 300. R is the rectangular region with vertices (0, 1), (0, 3), (3, 3), and (3, 1); (x,y)=x2y.In the following exercises, the region R occupied by a lamina is shown in a graph. Find the mass of R with the density function . 301. R is the trapezoidal region determined by the lines y=14x+52,y=0,y=2 . and x=0 ; (x,y)=3xy .In the following exercises, the region R occupied by a lamina is shown in a graph. Find the mass of R with the density function . 302. R is the trapezoidal region determined by the lines y=0,y=x,and y=x+3;(x,y)=2x+y.In the following exercises, the region R occupied by a lamina is shown in a graph. Find the mass of R with the density function . 303. R is the disk of radius 2 centered at (1 2): (x,y)=x2+y22x4y+5.In the following exercises, the region R occupied by a lamina is shown in a graph. Find the mass of R with the density function . 304. R is the unit disk; (x,y)=3x4+6x2y2+3y4.In the following exercises, the region R occupied by a lamina is shown in a graph. Find the mass of R with the density function . 305. R is the region enclosed by the ellipse x2+4y2=1;(x,y)=1.In the following exercises, the region R occupied by a lamina is shown in a graph. Find the mass of R with the density function . 306. R={(x,y)9x2+y21,x0,y0};(x,y)=9x2+y2.In the following exercises, the region R occupied by a lamina is shown in a graph. Find the mass of R with the density function . 307. R is the region bounded by y=x,y=xy=x+2;(x,y)=1.In the following exercises, the region R occupied by a lamina is shown in a graph. Find the mass of R with the density function . 308. 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 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. 309. [T] R is the triangular region with vertices (0, 0), (0. 3), and (6. 0): p(x. y) = xy.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. 310. [T] R is the triangular region 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 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. 311. [T] R is the rectangular region with vertices (0,0), (0,3), (6,3), and (6,0); (x,y)=xy 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. 312. [T] 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 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. 313. [T] R is the trapezoidal region determined by the lines y=14x+52,y=0, y=2 and x = 0: y=x+3;(x,y)=3xyp(x. y) = 3xy.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. 314. [T] R is the trapezoidal region determined by lines y=x+3;(x,y)=2x+y ,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. 315. [T] R is the disk of radius 2 centered at (1, 2): p(x.y)=x2+y2— 2x—4y+5.

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