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All Textbook Solutions for Calculus: Early Transcendentals (3rd Edition)
Regions of integration Write an iterated integral of a continuous function f over the region R. Use the order dy dx. Start by sketching the region of integration if it is not supplied.
40. R is the region in quadrants 2 and 3 bounded by the semicircle with radius 3 centered at (0, 0).
Regions of integration Write an iterated integral of a continuous function f over the region R. Use the order dy dx. Start by sketching the region of integration if it is not supplied.
41. R is the region bounded by the triangle with vertices (0, 0), (2, 0), and (1, 1).
Regions of integration Write an iterated integral of a continuous function f over the region R. Use the order dy dx. Start by sketching the region of integration if it is not supplied.
42. R is the region in the first quadrant bounded by the x-axis, the line x = 6 – y, and the curve .
Evaluating integrals Evaluate the following integrals. A sketch is helpful. 27.RxydA;R is bounded by x = 0, y = 2x + 1, and y = 2x + 5Evaluating integrals Evaluate the following integrals. A sketch is helpful. 28.R(x+y)dA;R is the region in the first quadrant bounded by x = 0, y = x2, and y = 8 x2.Evaluating integrals Evaluate the following integrals. A sketch is helpful. 29.Ry2dA;R is bounded by x = 1, y = 2x + 2, and y = x 1Evaluating integrals Evaluate the following integrals. A sketch is helpful. 30.Rx2ydA; R is the region in quadrants 1 and 4 bounded by the semicircle of radius 4 centered at (0, 0).Evaluating integrals Evaluate the following integrals. A sketch is helpful. 47.R12ydA; R is bounded by y = 2 x,y=x and y = 0.Evaluating integrals Evaluate the following integrals. A sketch is helpful. 48.Ry2dA; R is bounded by y = 1, y = 1 x, and y = x 1.Evaluating integrals Evaluate the following integrals. A sketch is helpful. 49.R3xydA; R is bounded by y = 2 x, y = 0, and x = 4 y2 in the first quadrant.Evaluating integrals Evaluate the following integrals. A sketch is helpful. 50.R(x+y)dA; R is bounded by y=|x| and y = 4.Evaluating integrals Evaluate the following integrals. A sketch is helpful. 51.R3x2dA; R is bounded by y = 0, y = 2x + 4, and y = x3.52EEvaluating integrals Evaluate the following integrals. A sketch is helpful. 53. R(x+y)dA; R is the region bounded by y = 1/x and y = 5/2 x.Evaluating integrals Evaluate the following integrals. A sketch is helpful. 54. Ry1+x+y2dA;R=(x,y):0xy,0y1Evaluating integrals Evaluate the following integrals. A sketch is helpful. 55. Rxsec2ydA;R=(x,y):0yx2,0x/2Evaluating integrals Evaluate the following integrals. A sketch is helpful. 56.R8xy1+x2+y2dA;R=(x,y):0yx,0x2Changing order of integration Reverse the order of integration in the following integrals. 57.02x22xf(x,y)dydxChanging order of integration Reverse the order of integration in the following integrals. 58.03062xf(x,y)dydxChanging order of integration Reverse the order of integration in the following integrals. 59.1/210lnyf(x,y)dxdyChanging order of integration Reverse the order of integration in the following integrals. 60.011ef(x,y)dxdyChanging order of integration Reverse the order of integration in the following integrals. 61.010cos1yf(x,y)dxdyChanging order of integration Reverse the order of integration in the following integrals. 62.1e0lnxf(x,y)dydxChanging order of integration Reverse the order of integration and evaluate the integral. 63. 01y1ex2dxdyChanging order of integration Reverse the order of integration and evaluate the integral. 64.0xsiny2dydxChanging order of integration Reverse the order of integration and evaluate the integral. 65.01/2y21/4ycos(16x2)dxdyChanging order of integration Reverse the order of integration and evaluate the integral. 66. 04x2xy5+1dydxChanging order of integration Reverse the order of integration and evaluate the integral. 67. 03y3x4cos(x2y)dxdyChanging order of integration Reverse the order of integration and evaluate the integral. 68. 0204x2xe2y4ydydxTwo integrals to one Draw the regions of integration and write the following integrals as a single iterated integral: 01eyef(x,y)dxdy+10eyef(x,y)dxdy.Two integrals to one Draw the regions of integration and write the following integrals as a single iterated integral. 70. 40016x2f(x,y)dydx+0404xf(x,y)dydxVolumes Find the volume of the following solids. 71. The solid bounded by the cylinder z = 2 y2, the xy-plane, the xz-plane, and the planes y = x and x =1Volumes Find the volume of the following solids. 72. The solid bounded between the cylinder z = 2sin2x and the xy-plane over the region R = {(x, y):0 x y }Volumes Use double integrals to calculate the volume of the following regions. 53.The tetrahedron bounded by the coordinate planes (x = 0, y = 0, z = 0) and the plane z = 8 2x 4yVolumes Use double integrals to calculate the volume of the following regions. 54.The solid in the first octant bounded by the coordinate planes and the surface z = 1 y x2Volumes Use double integrals to calculate the volume of the following regions. 55.The segment of the cylinder x2 + y2 = 1 bounded above plane z = 12 + x + y and below by z = 0Regions between surfaces Find the volume of the following solid regions. 74.The solid S between the surfaces z = ex y and z = ex y where S intersects the xy-plane in the region R = {(x, y): 0 x y, 0 y 1}Regions between surfaces Find the volume of the following solid regions. 73.The solid above the region R = {(x, y): 0 x 1, 0 y 2 x} and between the planes 4x 4y + z = 0 and 2x y + z = 8Volumes Find the volume of the following solids. 78. The solid in the first octant bounded by the planes x = 0, y = 0, z = 1, and z = y, and the cylinder y = 1 x2Regions between surfaces Find the volume of the following solid regions. 69.The solid above the region R = {(x, y): 0 x 1, 0 y 1 x} bounded by the paraboloids z = x2 + y2 and z = 2 x2 y2, and the coordinate planes in the first octantRegions between surfaces Find the volume of the following solid regions. 72.The solid bounded by the parabolic cylinder z = x2 + 1, and the planes z = y + 1 and y = 1Volume using technology Find the volume of the following solids. Use a computer algebra system to evaluate an appropriate iterated integral. 81. The column with a square base R = {(x, y): |x| 1, |y| 1} cut by the plane z = 4 x yVolume using technology Find the volume of the following solids. Use a computer algebra system to evaluate an appropriate iterated integral. 82. The solid between the paraboloid z = x2 + y2 and the plane z = 1 2yVolume using technology Find the volume of the following solids. Use a computer algebra system to evaluate an appropriate iterated integral. 83. The wedge sliced from the cylinder x2 + y2 = 1 by the planes z = a (2 x) and z = a(x 2), where a 0Volume using technology Find the volume of the following solids. Use a computer algebra system to evaluate an appropriate iterated integral. 84. The solid bounded by the elliptical cylinder x2 + 3y2 = 12, the plane z = 0, and the paraboloid z = 3x2 + y2 + 1Area of plane regions Use double integrals to compute the area of the following regions. 85. The region bounded by the parabola y = x2 and the line y = 4Area of plane regions Use double integrals to compute the area of the following regions. 86. The region bounded by the parabola y = x2 and the line y = x + 2Area of plane regions Use double integrals to compute the area of the following regions. 87. The region in the first quadrant bounded by y = ex and x = ln 2Area of plane regions Use double integrals to compute the area of the following regions. 88. The region bounded by y = 1 + sin x and y = 1 sin x on the interval [0, ]Area of plane regions Use double integrals to compute the area of the following regions. 89. The region in the first quadrant bounded by y = x2, y = 5x + 6, and y = 6 xArea of plane regions Use double integrals to compute the area of the following regions. 90. The region bounded by the lines x = 0, x = 4, y = x, and y = 2x + 1Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a.In the iterated integral cdabf(x,y)dxdy, the limits a and b must be constants or functions of x. b.In the iterated integral, cdabf(x,y)dxdy, the limits c and d must be functions of y. c.Changing the order of integration gives 021yf(x,y)dxdy=1y02f(x,y)dydx.Related integrals Evaluate each integral a. 0404(4xy)dxdy b. 0404|4xy|dxdyVolumes Compute the volume of the following solids. 97.Sliced block The solid bounded by the planes x = 0, x = 5, z = y 1, z = 2y 1, z = 0, and z = 2Square region Consider the region R = {(x, y): |x| + |y| 1} shown in the figure. a.Use a double integral to verify that the area of R is 2. b.Find the volume of the square column whose base is R and whose upper surface is z = 12 3x 4y. c.Find the volume of the solid above R and beneath the cylinder x2 + z2 = 1. d.Find the volume of the pyramid whose base is R and whose vertex is on the z-axis at (0, 0, 6).Average value Use the definition for the average value of a function over a region R(Section 16.1), f=1areaofRRf(x,y)dA. 95. Find the average value of a x y over the region R=(x,y):x+ya,x0,y0, where a 0Average value Use the definition for the average value of a function over a region R(Section 16.1) f=1areaofRRf(x,y)dA. 96. Find the average value of z = a2 x2 y2 over the region R=(x,y):x2+y2a2,where a 0Area integrals Consider the following regions R. a.Sketch the region R. b.Evaluate RdA to determine the area of the region. c.Evaluate RxydA. 91. R is the region between both branches of y = 1/x and the lines y = x + 3/2 and y = x 3/2.Area integrals Consider the following regions R. a.Sketch the region R. b.Evaluate RdA to determine the area of the region. c.Evaluate RxydA. 92. R is the region bounded by the ellipse x2/18 + y2/36 = 1 with y = 4x/3.Improper integrals Many improper double integrals may be handled using the techniques for improper integrals in one variable (Section 8.9). For example, under suitable conditions on f, agh(x)f(x,y)dydx=limxabg(x)h(x)f(x,y)dydx Use or extend the one-variable methods for improper integrals to evaluate the following integrals. 99. 10eyxydydx100EImproper integrals Many improper double integrals may be handled using the techniques for improper integrals in one variable (Section 8.9). For example, under suitable conditions on f, agh(x)f(x,y)dydx=limxabg(x)h(x)f(x,y)dydx Use or extend the one-variable methods for improper integrals to evaluate the following integrals. 101. 00exydydx102EDescribe in polar coordinates the region in the first quadrant between the circles of radius 1 and 2.Express the functions f(x, y) = (x2 + y2)5/2 and h(x, y) = x2 y2 in polar coordinates.Give a geometric explanation for the extraneous root z = 4 found in Example 2.
Example 2 Region bounded by two surfaces
Find the volume of the region bounded by the paraboloid z = x2 + y2 and the cone .
Express the area of the disk R ={(r, ) : 0 r a, 0 27} in terms of a double integral in polar coordinates.Draw the region {(r, ): 1 r 2, 0 /2}. Why is it called a polar rectangle?Write the double integral Rf(x,y)dAas an iterated integral in polar coordinates when R = {(r, ): a r b, }.Sketch in the xy-plane the region of integration for the integral
4EHow do you find the area of a region R = {(r, ): 0 g( ) r h(), }?How do you find the average value of a function over a region that is expressed in polar coordinates?Polar rectangles Sketch the following polar rectangles. 7.R = {(r, ): 0 r 5, 0 /2}Polar rectangles Sketch the following polar rectangles. 8.R = {(r, ): 2 r 3, /4 5/4}Polar rectangles Sketch the following polar rectangles. 9.R = {(r, ): 1 r 4, /4 2/3}Polar rectangles Sketch the following polar rectangles. 10.R = {(r, ): 4 r 5, /3 /2}Volume of solids Find the volume of the solid bounded by the surface z = f(x, y) and the xy-plane. 11. f(x,y)=4x2+y2Volume of solids Find the volume of the solid bounded by the surface z = f(x, y) and the xy-plane. 12. f(x, y) = 16 4(x2 + y2)Volume of solids Find the volume of the solid bounded by the surface z = f(x, y) and the xy-plane. 13. f(x,y)=e(x2+y2)/8e2Volume of solids Find the volume of the solid bounded by the surface z = f(x, y) and the xy-plane. 14. f(x,y)=201+x2+y22Solids bounded by paraboloids Find the volume of the solid below the paraboloid z = 4 x2 y2 and above the following regions. 11.R = {(r, ): 0 r 1, 0 2}Solids bounded by paraboloids Find the volume of the solid below the paraboloid z = 4 x2 y2 and above the following regions. 12.R = {(r, ): 0 r 2, 0 2}Solids bounded by paraboloids Find the volume of the solid below the paraboloid z = 4 x2 y2 and above the following regions. 13.R = {(r, ): 1 r 2, 0 2}Solids bounded by paraboloids Find the volume of the solid below the paraboloid z = 4 x2 y2 and above the following regions. 14.R = {(r, ): 1 r 2, /2 /2}Solids bounded by hyperboloids Find the volume of the solid below the hyperboloid z=51+x2+y2 and above the following regions. 17.R = {(r, ): 3 r 22, 0 2}Solids bounded by hyperboloids Find the volume of the solid below the hyperboloid z=51+x2+y2 and above the following regions. 18.R = {(r, ): 3 r 15, /2 2}Cartesian to polar coordinates Sketch the given region of integration R and evaluate the integral over R using polar coordinates. 23.R(x2+y2)dA; R = {(r, ): 0 r 4, 0 2}Cartesian to polar coordinates Sketch the given region of integration R and evaluate the integral over R using polar coordinates. 24.R2xydA; R = {(r, ): 1 r 3, 0 /2}Cartesian to polar coordinates Sketch the given region of integration R and evaluate the integral over R using polar coordinates. 23.R2xydA; R = {(x, y): x2 + y2 9, y 0}Cartesian to polar coordinates Sketch the given region of integration R and evaluate the integral over R using polar coordinates. 26.RdA1+x2+y2; R = {(r, ): 1 r 2, 0 }Cartesian to polar coordinates Sketch the given region of integration R and evaluate the integral over R using polar coordinates. 27.RdA16x2y2; R = {(x, y): x2 + y2 4, x 0, y 0}Cartesian to polar coordinates Sketch the given region of integration R and evaluate the integral over R using polar coordinates. 28.Rex2y2dA; R = {(x, y): x2 + y2 9}Cartesian to polar coordinates Evaluate the following integrals using polar coordinates. Assume (r, ) are polar coordinates. A sketch is helpful. 27. 111x21x2(x2+y2)3/2dydxCartesian to polar coordinates Evaluate the following integrals using polar coordinates. Assume (r, ) are polar coordinates. A sketch is helpful. 28. 0309x2x2+y2dydxCartesian to polar coordinates Evaluate the following integrals using polar coordinates. Assume (r, ) are polar coordinates. A sketch is helpful. 29. Rx2+y2dA;R=(x,y):1x2+y24Cartesian to polar coordinates Evaluate the following integrals using polar coordinates. Assume (r, ) are polar coordinates. A sketch is helpful. 30. 44016y2(16x2y2)dxdyRegions between surfaces Find the volume of the following solid regions. 71.The solid bounded by the paraboloid z = x2 + y2: and the plane z = 9Volume between surfaces Find the volume of the following solids. 21.The solid bounded by the paraboloid z = 2 x2 y2 and the plane z = 1Volume between surfaces Find the volume of the following solids. 19.The solid bounded by the paraboloids z = x2 + y2 and z = 2 x2 y2Volume between surfaces Find the volume of the following solids. 20.The solid bounded by the paraboloids z = 2x2 + y2 and z = 27 x2 2y2Volume between surfaces Find the volume of the following solids. 35. The solid bounded below by the paraboloid z = x2 + y2 x y and above by the plane x + y + z = 4Volume between surfaces Find the volume of the following solids. 36. The solid bounded by the cylinder x2 + y2 = 4 and the planes z = 3 x and z = x 3Volume between surfaces Find the volume of the following solids. 37. The solid bounded by the paraboloid z = 18 x2 3y2 and the hyperbolic paraboloid z = x2 y2Volume between surfaces Find the volume of the following solids. 38. The solid outside the cylinder x 2 + y2 1 that is bounded above by the hyperbolic paraboloid z = x2 + y2 + R and below by the paraboloid z = x2 + 3y2Volume between surfaces Find the volume of the following solids. 39. The solid outside the cylinder x2 + y2 = 1 that is bounded above by the sphere x2 + y2 + z2 = 8 and below by the cone z=x2+y2Volume between surfaces Find the volume of the following solids. 40. The solid bounded by the cone z = 2x2+y2 and the upper half of a hyperboloid of two sheets z=1+x2+y2Describing general regions Sketch the following regions R. Then express Rg(r,)dA as an iterated integral over R in polar coordinates. 41. The region inside the limaon r=1+12cosDescribing general regions Sketch the following regions R. Then express Rg(r,)dA as an iterated integral over R in polar coordinates. 42. The region inside the leaf of the rose r = 2 sin 2 in the first quadrantDescribing general regions Sketch the following regions R. Then express Rg(r,)dA as an iterated integral over R in polar coordinates. 43. The region inside the lobe of the lemniscates r2 = 2 sin 2 in the first quadrantDescribing general regions Sketch the following regions R. Then express Rg(r,)dA as an iterated integral over R in polar coordinates. 44. The region outside the circle r = 2 and inside the circ le r = 4 sinDescribing general regions Sketch the following regions R. Then express Rg(r,)dA as an iterated integral over R in polar coordinates. 45. The region outside the circle r = 1 and inside the rose r = 2 sin 3 in the first quadrantDescribing general regions Sketch the following regions R. Then express Rg(r,)dA as an iterated integral over R in polar coordinates. 46. The region outside the circle r = 1/2 and inside the cardioids r = 1 + cosComputing areas Use a double integral to find the area of the following regions. 47. The annular region {(r, ): 1 r 2, 0Computing areas Use a double integral to find the area of the following regions. 48. The region bounded by the cardioid r = 2(1 sin )Computing areas Use a double integral to find the area of the following regions. 49. The region bounded by all leaves of the rose r = 2 cos 3Computing areas Use a double integral to find the area of the following regions. 50. The region inside both the cardioid r = 1 cos and the circle r = 147-52. Computing areas Use a double integral to find the area of the following regions.
51. The region inside both the cardioid r = 1 + sin θ and the cardioid r = 1 + cos θ
Computing areas Use a double integral to find the area of the following regions. 52. The region bounded by the spiral r = 2, for 0 , and the x-axisAverage values Find the following average values. 45.The average distance between points of the disk {(r, ): 0 r a} and the originAverage values Find the following average values. 48.The average value of 1/r2 over the annulus {(r, ): 2 r 4}Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a.Let R be the unit disk centered at (0,0). Then R(x2+y2)dA=0201r2drd. b.The average distance between the points of the hemisphere z=4x2y2 and the origin is 2 (calculus not required). c.The integral 0101y2ex2+y2dxdy is easier to evaluate in polar coordinates than in Cartesian coordinates.Areas of circles Use integration to show that the circles r = 2a cos and r = 2a sin have the same area, which is a2.Filling bowls with water Which bowl holds more water if it is filled to a depth of 4 units? The paraboloid z = x2 + y2, for 0 z 4 The cone z=x2+y2, for 0 z 4 The hyperboloid z=1+x2+y2, for 1 z 5Equal volumes To what height (above the bottom of the bowl) must the cone and paraboloid bowls of Exercise 57 be filled to hold the same volume of water as the hyperboloid bowl filled to a depth of 4 units (1 ≤ z ≤ 5)?
Volume of a hyperbolic paraboloid Consider the surface z = x2 y2. a.Find the region in the xy-plane in polar coordinates for which z 0. b.Let R = {(r, ): 0 r a, /4 /4} , which is a sector of a circle of radius a. Find the volume of the region below the hyperbolic paraboloid and above the region R.Volume of a sphere Use double integrals in polar coordinates to verify that the volume of a sphere of radius a is 43a3.Volume Find the volume of the solid bounded by the cylinder (x 1)2 + y2 = 1, the plane z = 0, and the cone z=x2+y2 (see figure). (Hint: Use symmetry.)Volume Find the volume of the solid bounded by the paraboloid z = 2x2 + 2y2, the plane z = 0, and the cylinder x2 + (y 1)2 = 1. (Hint: Use symmetry.)Miscellaneous integrals Evaluate the following integrals using the method of your choice. A sketch is helpful. 57.RdA4+x2+y2;R={(r,):0r2,/23/2}Miscellaneous integrals Evaluate the following integrals using the method of your choice. A sketch is helpful. 56.Rxyx2+y2+1dA; R is the region bounded by the unit circle centered at the origin.Improper integrals Improper integrals arise in polar coordinates when the radial coordinate r becomes arbitrarily large. Under certain conditions, these integrals are treated in the usual way: af(r,)rdrd=limbabf(r,)rdrd. Use this technique to evaluate the following integrals. 63.0/21cosr3rdrdImproper integrals Improper integrals arise in polar coordinates when the radial coordinate r becomes arbitrarily large. Under certain conditions, these integrals are treated in the usual way: af(r,)rdrd=limbabf(r,)rdrd. Use this technique to evaluate the following integrals. 64.RdA(x2+y2)5/2;R={(r,):1r,02}Improper integrals Improper integrals arise in polar coordinates when the radial coordinate r becomes arbitrarily large. Under certain conditions, these integrals are treated in the usual way: af(r,)rdrd=limbabf(r,)rdrd. Use this technique to evaluate the following integrals. 65.Rex2y2dA;R={(r,):0r,0/2}Improper integrals Improper integrals arise in polar coordinates when the radial coordinate r becomes arbitrarily large. Under certain conditions, these integrals are treated in the usual way: af(r,)rdrd=limbabf(r,)rdrd. Use this technique to evaluate the following integrals. 66.RdA(1+x2+y2)2; R is the first quadrant.69EMass from density data The following table gives the density (in units of g/cm2) at selected points of a thin semicircular plate of radius 3. Estimate the mass of the plate and explain your method.A mass calculation Suppose the density of a thin plate represented by the region R is (r, ) (in units of mass per area). The mass of the plate is R(r,)dA. Find the mass of the thin half annulus R = {(r, ): 1 r 4,0 } with a density (r, ) = 4 + r sin .Area formula In Section 12.3 it was shown that the area of a region enclosed by the polar curve r = g() and the rays = and = , where 2, is A=12r2d. Prove this result using the area formula with double integrals.Normal distribution An important integral in statistics associated with the normal distribution is I=ex2dx. It is evaluated in the following steps. a. In Section 8.9, it is shown that 0ex2dx converges (in the narrative following Example 7). Use this result to explain why ex2dx converges. b. Assume I2=(ex2dx)(ey2dx)=ex2y2dxdy, where we have chosen the variables of integration to be x and y and then written the product as an iterated integral. Evaluate this integral in polar coordinates and show that I=. Why is the solution I= rejected? c. Evaluate 0ex2dx, 0xex2dx, and 0x2ex2dx (using part (a) if needed).Existence of integrals For what values of p does the integral RdA(x2+y2)p exist in the following cases? a.R={(r,):1r,02} b.R={(r,):0r1,02}Integrals in strips Consider the integral I=RdA(1+x2+y2)2, where R = {(x, y): 0 x 1, 0 y a}. a.Evaluate I for a = 1. (Hint: Use polar coordinates.) b.Evaluate I for arbitrary a 0. c.Let a in part (b) to find I over the infinite strip R = {(x, y): 0 x 1, 0 y }.List the six orders in which the three differentials dx, dy, and dz may be written.Write the integral in Example 1 in the orders dx dy dz and dx dz dy.Write the integral in Example 2 in the orders dz dy dx and dx dy dz.Without integrating, what is the average value of f(x, y, z) = sin x sin y sin z on the cube {(x, y, z): 1 x 1, 1 y 1, 1 z 1}? Use symmetry arguments.Sketch the region D = {(x, y, z): x2 + y2 4, 0 z 4}.Write an iterated integral for Df(x,y,z)dV, where D is the box {(x, y, z): 0 x 3, 0 y 6, 0 z 4}.Write an iterated integral for Df(x,y,z)dV, where D is a sphere of radius 9 centered at (0,0,0). Use the order dz dy dx.Sketch the region of integration for the integral 0101z201y2z2f(x,y,z)dxdydz.Write the integral in Exercise 4 in the order dy dx dz. 4.Sketch the region of integration for the integral 0101z201y2z2f(x,y,z)dxdydz.Write an integral for the average value of f(x, y, z) = xyz over the region bounded by the paraboloid z = 9 x2 y2 and the xy-plane (assuming the volume of the region is known).Integrals over boxes Evaluate the following integrals. A sketch of the region of integration may be useful. 7.223602dxdydzIntegrals over boxes Evaluate the following integrals. A sketch of the region of integration may be useful. 8.1112016xyzdydxdzIntegrals over boxes Evaluate the following integrals. A sketch of the region of integration may be useful. 9.22121exy2zdzdxdyIntegrals over boxes Evaluate the following integrals. A sketch of the region of integration may be useful. 10.0ln40ln30ln2ex+y+zdxdydzIntegrals over boxes Evaluate the following integrals. A sketch of the region of integration may be useful. 11.0/2010/2sinxcosysin2zdydxdzIntegrals over boxes Evaluate the following integrals. A sketch of the region of integration may be useful. 12.021201yzexdxdzdyIntegrals over boxes Evaluate the following integrals. A sketch of the region of integration may be useful. 13.D(xy+xz+yz)dV;D={(x,y,z):1x1,2y2,3z3}Integrals over boxes Evaluate the following integrals. A sketch of the region of integration may be useful. 14.Dxyzex2y2dV;D={(x,y,z):0xln2,0yln4,0z1}Volumes of solids. Find the volume of the following solids using triple integrals. 15.The solid in the first octant bounded by the plane 2x + 3y + 6z = 12 and the coordinate planesVolumes of solids. Find the volume of the following solids using triple integrals. 16.The solid in the first octant formed when the cylinder z = sin y, for 0 y , is sliced by the planes y = x and x = 0Volumes of solids. Find the volume of the following solids using triple integrals. 19.The wedge above the xy-plane formed when the cylinder x2 + y2 = 4 is cut by the planes z = 0 and y = zVolumes of solids. Find the volume of the following solids using triple integrals. 18.The prism in the first octant bounded by z = 2 4x and y = 8Volumes of solids. Find the volume of the following solids using triple integrals. 22.The solid bounded by the surfaces z = ey and z = 1 over the rectangle {(x, y): 0 x 1, 0 y ln 2}.Volumes of solids. Find the volume of the following solids using triple integrals. 20.The wedge bounded by the parabolic cylinder y = x2 and the planes z = 3 y and z = 0Volumes of solids. Find the volume of the following solids using triple integrals. 21.The solid between the sphere x2 + y2 + z2 = 19 and the hyperboloid z2 x2 y2 = l, for z 0Volumes of solids. Find the volume of the following solids using triple integrals. 17.The solid bounded below by the cone z=x2+y2 and bounded above by the sphere x2 + y2 + z2 = 8Volumes of solids Use a triple integral to find the volume of the following solids. 23. The solid bounded by the cylinder y = 9 x2 and the paraboloid y = 2x2 + 3z2Volumes of solids Use a triple integral to find the volume of the following solids. 24. The wedge in the first octant bounded by the cylinder x = z2 and the planes z = 2 x, y = 2, and z = 0Volumes of solids Use a triple integral to find the volume of the following solids. 25. The wedge of the cylinder x2 + 4z2 = 4 created by the planes y = 3 x and y = x 3Finding an appropriate order of integration Find the volume of the following solids. 36.The solid bounded by x = 0, x = 2, y = 0, y = ez, z = 0, and z = 1Finding an appropriate order of integration Find the volume of the following solids. 35.The solid bounded by x = 0, x = 1 z2, y = 0, z = 0, and z = 1 yFinding an appropriate order of integration Find the volume of the following solids. 38.The solid bounded by x = 0, y = z2, z = 0, and z = 2 x yFinding an appropriate order of integration Find the volume of the following solids. 37.The solid bounded by x = 0, x = 2, y = z, y = z + l, z = 0, and z = 4Six orderings Let D be the solid in the first octant bounded by the planes y = 0 and y = x, and the cylinder 4x2 + z2 = 4. Write the triple integral of f(x, y, z) over D in the given order of integration 30. dz dy dxSix orderings Let D be the solid in the first octant bounded by the planes y = 0 and y = x, and the cylinder 4x2 + z2 = 4. Write the triple integral of f(x, y, z) over D in the given order of integration 31. dz dx dySix orderings Let D be the solid in the first octant bounded by the planes y = 0 and y = x, and the cylinder 4x2 + z2 = 4. Write the triple integral of f(x, y, z) over D in the given order of integration 32. dy dx dzSix orderings Let D be the solid in the first octant bounded by the planes y = 0 and y = x, and the cylinder 4x2 + z2 = 4. Write the triple integral of f(x, y, z) over D in the given order of integration 33. dy dz dxSix orderings Let D be the solid in the first octant bounded by the planes y = 0 and y = x, and the cylinder 4x2 + z2 = 4. Write the triple integral of f(x, y, z) over D in the given order of integration 34. dx dy dzSix orderings Let D be the solid in the first octant bounded by the planes y = 0, z = 0, and y = x, and the cylinder 4x2 + z2 4. Write the triple integral of f(x, y. z) over D in the given order of integration.
dx dz dy
All six orders Let D be the solid bounded by y = x, z = 1 y2, x = 0, and z = 0. Write triple integrals over D in all six possible orders of integration.Changing order of integration Write the integral 020101ydzdydx in the five other possible orders of integration.Triple integrals Evaluate the following integrals. 30.000sinxsinydzdxdyTriple integrals Evaluate the following integrals. 33.0204y24xdzdxdyTriple integrals Evaluate the following integrals. 26.0101x201x2y22xzdzdydxTriple integrals Evaluate the following integrals. 25.0101x201x2dzdydxTriple integrals Evaluate the following integrals. 28.16042y/30122y3z1ydxdzdyTriple integrals Evaluate the following integrals. 29.0309z201+x2+z2dydxdzTriple integrals Evaluate the following integrals. 44. 01y2y02xy15xydzdxdyTriple integrals Evaluate the following integrals. 31.1ln81zlnyln2yex+y2zdxdydzTriple integrals Evaluate the following integrals. 32.0101x202x4yzdzdydxChanging the order of integration Rewrite the following integrals using the indicated order of integration and then evaluate the resulting integral. 39.051004x+4dydxdz in the order dz dx dyChanging the order of integration Rewrite the following integrals using the indicated order of integration and then evaluate the resulting integral. 40.012204y2dzdydx in the order dy dz dxChanging the order of integration Rewrite the following integrals using the indicated order of integration and then evaluate the resulting integral. 41.0101x201x2dydzdx in the order dz dy dxChanging the order of integration Rewrite the following integrals using the indicated order of integration and then evaluate the resulting integral. 42.04016x2016x2z2dydzdx in the order dz dy dxAverage value Find the following average values. 51. The average value of f(x, y, z) = 8xy cos z over the points inside the box D = {(x, y, z): 0 x 1, 0 y 2, 0 z /2Average value Find the following average values. 43.The average temperature in the box D = {(x, y, z): 0 x ln 2, 0 y ln 4, 0 z ln 8} with a temperature distribution of T(x, y, z) = 128 exyzAverage value Find the following average values. 45.The average of the squared distance between the origin and points in the solid cylinder D = {(x, y, z): x2 + y2 4, 0 z 2}Average value Find the following average values. 47.The average z-coordinate of points on and within a hemisphere of radius 4 centered at the origin with its base in the xy-planeExplain why or why not Determine whether the following statements are true and give an explanation or counterexample. a.An iterated integral of a function over the box D = {(x, y, z): 0 x a, 0 y b, 0 z c} can be expressed in eight different ways. b.One possible iterated integral of f over the prism D = {(x, y, z): 0 x 1, 0 y 3x 3, 0 z 5} is 03x30105f(x,y,z)dzdxdy. c.The region D={(x,y,z):0x1,0y1x2,0z1x2} is a sphere.Changing the order of integration Use another order of integration to evaluate 14z4z02sinyzx3/2dydxdz.57EMiscellaneous volumes Use a triple integral to compute the volume of the following regions. 54.The solid common to the cylinders z = sin x and z = sin y over the square R = {(x, y): 0 x , 0 y } (The figure shows the cylinders, but not the common region.)59E60EMiscellaneous volumes Use a triple integral to compute the volume of the following regions. 53.The pyramid with vertices (0, 0, 0), (2, 0, 0), (2, 2, 0), (0, 2, 0), and (0, 0, 4)Volumes of solids. Find the volume of the following solids using triple integrals. 24.The solid in the first octant bounded by the cone z=1x2+y2 and the plane x + y + z = 1Two cylinders The x- and y-axes form the axes of two right circular cylinders with radius 1 (see figure). Find the volume of the solid that is common to the two cylinders.Three cylinders The coordinate axes form the axes of three right circular cylinders with radius 1 (see figure). Find the volume of the solid that is common to the three cylinders.
Dividing the cheese Suppose a wedge of cheese fills the region in the first octant bounded by the planes y = z, y = 4, and x = 4. You could divide the wedge into two pieces of equal volume by slicing the wedge with the plane x = 2. Instead find a with 0 a 4 such that slicing the wedge with the plane y = a divides the wedge into two pieces of equal volume.Partitioning a cube Consider the region D1 = {(x, y, z): 0 x y z 1}. a. Find the volume of D1. b. Let D2, , D6 be the cousins of D1 formed by remaining x, y, and z in the inequality 0 x y z 1. Show that the volumes of D1,, D6 are equal. c. Show that the union of D1,, D6 is a unit cube.General volume formulas Find equations for the bounding surfaces, set up a volume integral, and evaluate the integral to obtain a volume formula for each region. Assume that a, b, c, r, R, and h are positive constants. 61.Cone Find the volume of a right circular cone with height h and base radius r.68EGeneral volume formulas Find equations for the bounding surfaces, set up a volume integral, and evaluate the integral to obtain a volume formula for each region. Assume that a, b, c, r, R, and h are positive constants. 63. Spherical cap Find the volume of the cap of a sphere of radius R with height h.70E71E72EHypervolume Find the Volume of the four-dimensional pyramid bounded by w + x + y + z + 1=0 and the coordinate planes w = 0, x = 0, y = 0, and z = 0.74E1QCFind the limits of integration for a triple integral in cylindrical coordinates that gives the volume of a cylinder with height 20 and a circular base of radius 10 centered at the origin in the xy-plane.Find the spherical coordinates of the point with rectangular coordinates (1, 3, 2). Find the rectangular coordinates of the point with spherical coordinates (2, /4, /4).Explain how cylindrical coordinates are used to describe a point in 3.Explain how spherical coordinates are used to describe a point in 3.Describe the set {(r, , z): r = 4z} in cylindrical coordinates.Describe the set {(, , ): = /4} in spherical coordinates.Explain why dz r dr d is the volume of a small box in cylindrical coordinates.Explain why 2 sin d d d is the volume of a small box in spherical coordinates.Write the integral D w(r, , z) dV as an iterated integral, where the region D, expressed in cylindrical coordinates, is D = {(r, , z) : G(r, ) z H(r, ), g() r h(), }.Write the integral D w(, , ) dV as an interated integral, where the region D, expressed in spherical coordinates, is D = {(, , ) : g(, ) h(, ), a b, }.What coordinate system is suggested if the integrand of a triple integral involves x2 + y2?What coordinate system is suggested if the integrand of a triple integral involves x2 + y2 + z2?Sets in cylindrical coordinates Identify and sketch the following sets in cylindrical coordinates. 11.{(r,,z):0r3,0/3,1z4}Sets in cylindrical coordinates Identify and sketch the following sets in cylindrical coordinates. 12.{(r,,z):0/2,z=1}Sets in cylindrical coordinates Identify and sketch the following sets in cylindrical coordinates. 13.{(r,,z):2rz4}Sets in cylindrical coordinates Identify and sketch the following sets in cylindrical coordinates. 14.{(r,,z):0z82r}Integrals in cylindrical coordinates Evaluate the following integrals in cylindrical coordinates. The figures illustrate the region of integration. 15.020111dzrdrdIntegrals in cylindrical coordinates Evaluate the following integrals in cylindrical coordinates. The figures illustrate the region of integration. 16.039y29y2093x2+y2dzdxdyIntegrals in cylindrical coordinates Evaluate the following integrals in cylindrical coordinates. The figures illustrate the region of integration. 17.111y21y211(x2+y2)3/2dzdxdyIntegrals in cylindrical coordinates Evaluate the following integrals in cylindrical coordinates. The figures illustrate the region of integration. 18.3309x20211+x2+y2dzdydxIntegrals in cylindrical coordinates Evaluate the following integrals in cylindrical coordinates. 19.0402/2x1x2ex2y2dydxdzIntegrals in cylindrical coordinates Evaluate the following integrals in cylindrical coordinates. 20.4416x216x2x2+y24dzdydzIntegrals in cylindrical coordinates Evaluate the following integrals in cylindrical coordinates. 21.0309x20x2+y2(x2+y2)1/2dzdydzIntegrals in cylindrical coordinates Evaluate the following integrals in cylindrical coordinates. 21.1101/23y1y2(x2+y2)1/2dxdydzMass from density Find the mass of the following objects with the given density functions. 23.The solid cylinder D={(r,,z):0r4,0z10} with density (r,,z)=1+z/2Mass from density Find the mass of the following objects with the given density functions. 24.The solid cylinder {(r,,z):0r3,0z2} with density (r,,z)=5er2Mass from density Find the mass of the following objects with the given density functions. 25.The solid cone D={(r,,z):0z6r,0r6} with density (r,,z)=7zMass from density Find the mass of the following objects with the given density functions. 26.The solid paraboloid {(r,,z):0z9r2,0r3} with (r,,z)=1+z/9Which weighs more? For 0 r 1, the solid bounded by the cone z = 4 4r and the solid bounded by the paraboloid z = 4 4r2 have the same base in the xy-plane and the same height. Which object has the greater mass if the density of both objects is (r, , z) = 10 2z?28EVolumes in cylindrical coordinates Use cylindrical coordinates to find the volume of the following solids. 29. The solid bounded by the plane z = 0 and the hyperboloid z=31+x2+y2Volumes in cylindrical coordinates Use cylindrical coordinates to find the volume of the following solids. 30.The solid bounded by the plane z = 25 and the paraboloid z = x2 + y2Volumes in cylindrical coordinates Use cylindrical coordinates to find the volume of the following solids. 31.The solid bounded by the plane z=29 and the hyperboloid z=4+x2+y2Volumes in cylindrical coordinates Use cylindrical coordinates to find the volume of the following solids. 32.The solid cylinder whose height is 4 and whose base is the disk {(r,):0r2cos}Volumes in cylindrical coordinates Use cylindrical coordinates to find the volume of the following solids. 33.The solid in the first octant bounded by the cylinder r = 1, and the planes z = x and z = 0Volumes in cylindrical coordinates Use cylindrical coordinates to find the volume of the following solids. 34.The solid bounded by the cylinders r = 1 and r = 2, and the planes z = 4 x y and z = 035ESets in spherical coordinates Identify and sketch the following sets in spherical coordinates. 36.{(,,):=2csc,0}Sets in spherical coordinates Identify and sketch the following sets in spherical coordinates. 37.{(,,):=4cos,0/2}Sets in spherical coordinates Identify and sketch the following sets in spherical coordinates. 38.{(,,):=2sec,0/2}Seattle has a latitude of 47.6° North and a longitude of 122.3° West; Rome, Italy, has a latitude of 41.9° North and a longitude of 12.5° East.
Find the approximate spherical and rectangular coordinates of Seattle. Express the angular coordinates in radians.
Find the approximate spherical and rectangular coordinates of Rome.
Consider the intersection curve of a sphere, and a plane passing through the center of the sphere and two points A and В on the sphere. It can be shown that the arc length of the segment of the intersection curve from A to B is the shortest distance on the sphere from A to B. Find the approximate shortest distance from Seattle to Rome. (Hint: Recall that , where 0 ≤ θ ≤ π is the angle between u and v; use the arc length formula s = rθ to find the distance.)
Los Angeles has a latitude of 34.05* North and a longitude of 118.24* West, and New York City has a latitude of 40.71* North and a longitude of 74.01* West. Find the approximate shortest distance from Los Angeles to New York City.
Integrals in spherical coordinates Evaluate the following integrals in spherical coordinates. 39.D(x2+y2+z2)5/2dV; D is the unit ball.Integrals in spherical coordinates Evaluate the following integrals in spherical coordinates. 40.De(x2+y2+z2)3/2dV; D is the unit ball.Integrals in spherical coordinates Evaluate the following integrals in spherical coordinates. 41.DdV(x2+y2+z2)3/2; D is the solid between the spheres of radius 1 and 2 centered at the origin.Integrals in spherical coordinates Evaluate the following integrals in spherical coordinates. 42.020/304sec2sindddIntegrals in spherical coordinates Evaluate the following integrals in spherical coordinates. 43.00/62sec42sindddIntegrals in spherical coordinates Evaluate the following integrals in spherical coordinates. 44.020/412sec(3)2sindddIntegrals in spherical coordinates Evaluate the following integrals in spherical coordinates. 45.02/6/302csc2sindddVolumes in spherical coordinates Use spherical coordinates to find the volume of the following solids. 46.A ball of radius a 0.Volumes in spherical coordinates Use spherical coordinates to find the volume of the following solids. 47.The solid bounded by the sphere = 2 cos and the hemisphere = 1, z 0Volumes in spherical coordinates Use spherical coordinates to find the volume of the following solids. 48.The solid cardioid of revolution D={(,,):01+cos,0,02}Volumes in spherical coordinates Use spherical coordinates to find the volume of the following solids. 49.The solid outside the cone =/4and inside the sphere = 4 cos ?Volumes in spherical coordinates Use spherical coordinates to find the volume of the following solids. 50.The solid bounded by the cylinders r = 1 and r = 2, and the cones = /6 and = /3Volumes in spherical coordinates Use spherical coordinates to find the volume of the following solids. 51.That part of the ball 4 that lies between the planes z = 2 and z=23Volumes in spherical coordinates Use spherical coordinates to find the volume of the following solids. 52.The solid inside the cone z = (x2 + y2)1/2 that lies between the planes z = 1 and z = 2Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. a.Any point on the z-axis has more than one representation in both cylindrical and spherical coordinates. b.The sets {(r, , z): r = z} and {(, , ): = /4} are the same.Spherical to rectangular Convert the equation 2 = sec 2, where 0 /4, to rectangular coordinates and identify the surface.Spherical to rectangular Convert the equation 2 = sec 2, where /4 /2, to rectangular coordinates and identify the surface.58EMass from density Find the mass of the following solids with the given density functions. Note that density is described by the function f to avoid confusion with the radial spherical coordinate . 57.The ball of radius 8 centered at the origin with a density f(, , ) = 2e3Mass from density Find the mass of the following solids with the given density functions. Note that density is described by the function f to avoid confusion with the radial spherical coordinate . 58.The solid cone {(r,,z):0z4,0r3z,02} with density f(r, , z) = 5 zMass from density Find the mass of the following solids with the given density functions. Note that density is described by the function f to avoid confusion with the radial spherical coordinate . 59.The solid cylinder {(r,,z):0r2,02,1z1} with a density of f(r, , z) = (2 |z|)(4 r)Changing order of integration If possible, write iterated integrals in cylindrical coordinates for the following regions in the specified orders. Sketch the region of integration. 60.The solid outside the cylinder r = 1 and inside the sphere = 5, for z 0, in the orders dz dr d, dr dz d, and d dz drChanging order of integration If possible, write iterated integrals in cylindrical coordinates for the following regions in the specified orders. Sketch the region of integration. 61.The solid above the cone z = r and below the sphere = 2, for z 0, in the orders dz dr d, dr dz d, and d dz drChanging order of integration if possible, write iterated integrals in spherical coordinates for the following regions in the specified orders. Sketch the region of integration. Assume g is continuous on the region. 64. 020/204secg(,,)2 sin d d d in the orders d d d and d d dChanging order of integration if possible, write iterated integrals in spherical coordinates for the following regions in the specified orders. Sketch the region of integration. Assume g is continuous on the region. 65. 02/6/2csc2g(,,)2 sin d d d in the orders d d d and d d dMiscellaneous volumes Choose the best coordinate system for finding the volume of the following solids. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables. 64.The solid inside the sphere = 1 and below the cone = /4, for z 0Miscellaneous volumes Choose the best coordinate system for finding the volume of the following solids. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables. 65.That part of the solid cylinder r 2 that lies between the cones = /3 and = 2/3Miscellaneous volumes Choose the best coordinate system for finding the volume of the following solids. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables. 66.That part of the ball 2 that lies between the cones = /3 and = 2/3Miscellaneous volumes Choose the best coordinate system for finding the volume of the following solids. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables. 67.The solid bounded by the cylinder r = 1, for 0 z x + yMiscellaneous volumes Choose the best coordinate system for finding the volume of the following solids. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables. 68.The solid inside the cylinder r = 2 cos , for 0 z 4 xMiscellaneous volumes Choose the best coordinate system for finding the volume of the following solids. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables. 69.The wedge cut from the cardioid cylinder r = 1 + cos by the planes z = 2 x and z = x 2Miscellaneous volumes Choose the best coordinate system for finding the volume of the following solids. Surfaces are specified using the coordinates that give the simplest description, but the simplest integration may be with respect to different variables. 70.Volume of a drilled hemisphere Find the volume of material remaining in a hemisphere of radius 2 after a cylindrical hole of radius 1 is drilled through the center of the hemisphere perpendicular to its base.Density distribution A right circular cylinder with height 8 cm and radius 2 cm is filled with water. A heated filament running along its axis produces a variable density in the water given by (r)=10.05e0.01r2g/cm3 ( stands for density here, not the radial spherical coordinate). Find the mass of the water in the cylinder. Neglect the volume of the filament.Charge distribution A spherical cloud of electric charge has a known charge density Q(), where is the spherical coordinate. Find the total charge in the interior of the cloud in the following cases. a.Q()=21044,1 b.Q()=(2104)e0.013,0Gravitational field due to spherical shell A point mass m is a distance d from the center of a thin spherical shell of mass M and radius R. The magnitude of the gravitational force on the point mass is given by the integral F(d)=GMm4020(dRcos)sin(R2+d22Rdcos)3/2dd, where G is the gravitational constant. a.Use the change of variable x = cos to evaluate the integral and show that if d R, then F(d)=GMmd2, which means the force is the same as if the mass of the shell were concentrated at its center. b.Show that if d R (the point mass is inside the shell), then F = 0.Water in a gas tank Before a gasoline-powered engine is started, water must be drained from the bottom of the fuel tank. Suppose the tank is a right circular cylinder on its side with a length of 2 ft and a radius of 1 ft. If the water level is 6 in above the lowest part of the tank, determine how much water must be drained from the tank.General volume formulas Use integration to find the volume of the following solids. In each case, choose a convenient coordinate system, find equations for the bounding surfaces, set up a triple integral, and evaluate the integral. Assume that a, b, c, r, R, and h are positive constants. 77.Cone Find the volume of a solid right circular cone with height h and base radius r.