I IS II. = 2y"\ V= n{ay 2 l 3 -\y 2 )dy. Jo n [fy 5 ' 3 1

Transcription

1 r Exercises 5.2 Figure 530 (a) EXAMPLE'S The region in the first quadrant bounded by the graphs of y = i* and y = 2x is revolved about the y-axis. Find the volume of the resulting solid. SOLUTON The region and a typical horizontal rectangle are shown in Figure 5.30(a). We wish to integrate with respect to y, so we solve the given equations for x in terms of y, obtaining = 2y"\ b and Figure 5.30(b) illustrates the volume generated by the region and the washer generated by the rectangle. We note the following: thickness of washer: dy outer radius: 2y 1 ' 3 inner radius: y volume: n[(2y 1/3 ) 2 - (±y) 2 ]dy = n(4y 2/3 - \y 2 ) dy i i i > Applying the limit of sums operator J 0 gives us the volume: f 8 V= n{ay 2 l 3 -\y 2 )dy. n [fy 5 ' ,31 512^. i2 y J 0 - -w EXERCSES 5.2 Exer. 1-4: Set up an integral that can be used to find the volume of the solid obtained by revolving the shaded region about the indicated axis. - -."J S i! 4 * a H l l\ ft:!

2 CHAPTER 5 Applications of the Definite ntegral ky + (3,4) * - V25 - y 2 + (3.-4) y = -2X x+y = l, x y = 1, x=2; y-axis 21 y = sin2x, x = 0, x = n, y = 0; x-axis (Hint: Use a half-angle formula.) 22 y = 1+cos3x, x = 0, x = 27T, y = 0; x-axis (Hint: Use a half-angle formula.) 23 y = sinx, y=cosx, x = 0, x = n/4; x-axis (Hint: Use a double angle formula.) 24 y = secx, y = sinx, x = 0, x = TT/4; x-axis Exer : Sketch the region R bounded by the graphs of the equations, and find the volume of the solid generated if R is revolved about the given line. 25 y = x 2, y = 4 (a)y=4 (b)y = 5 (c) x = 2 (d) x = 3 26 y = Vx. y = 0, x = 4 (a) x = 4 (b) x = 6 (c) y = 2 (d) y = 4 Exer : Set up an integral that can be used to find the volume of the solid generated by revolving the shaded region about the line (a) y = -2, (b) y = 5, (c) x = 7, and (d)x= ky Exer. 5-24: Sketch the region R bounded by the graphs of the equations, and find the volume of the solid generated if R is revolved about the indicated axis. 5 y = l/x, x = 1, x = 3, y = 0; x-axis 6 y = V^c. x = 4, y = 0; x-axis 7 y = x 2 4x, y = 0; x-axis 8 y = x 3, x = -2, y = 0; x-axis a y = x 2, y = 2; y-axis 0 y = l/x, y = l, y = 3, x = 0; y-axis 1 x = 4y y 2, x = 0; y-axis 2 y = x, y = 3, x = 0; y-axis 3 y = x 2, y = 4 x 2 ; x-axis 4 x = y 3, x 2 + y = 0; x-axis 5y=x, x + y = 4, x = 0; x-axis 6 y = (x - l) 2 + 1, y = -( x - 1)2 + 3; j.axjs 28 7 y 2 = x, 2y = x; y-axis 8 y = 2x, y = 4x 2 ; y-axis 9 x = y 2, x - y = 2; y-axis

3 430 CHAPTER Volume of a Lab Glass revolving the graph of r Applications of ntegration A glass container can be modeled by 2.2x * , VOlx* y = 2.95, 0 < x < < x < 15 about the x-axis, where x and y are measured in centimeters. Use a graphing utility to graph the function and find the volume of the container. 53. Find the volume of the solid generated if the.upper half of the ellipse 9x2 + 25y2 = 225 is revolved about 56. Modeling Data A draftsman is asked to determine the amount of material required to produce a machine part (see figure in first column). The diameters d of the part at equally spaced points x are listed in the table. The measurements are listed in centimeters. X d X d (a) the x-axis to form a prolate spheroid (shaped like a football). (b) the y-axis to form an oblate spheroid (shaped like half of a candy). y A 5- «5- -5 i - Figure for 53(a) (b) Use the regression capabilities of a graphing utility to find a fourth-degree polynomial through the points representing the radius of the solid. Plot the data and graph the model. (c) Use a graphing utility to approximate the definite integral yielding the volume of the part. Compare the result with the answer to part (a). Figure for 53(b) 54. Minimum Volume The arc of y = 4 (x2/4) on the interval [0, 4] is revolved about the line y b (see figure). (a) Find the volume of the resulting solid as a function of b. /~ (a) Use these data with Simpson's Rule to approximate the volume of the part. (b) Use a graphing utility to graph the function in part (a), and use the graph to approximate the value of b that minimizes the volume of the solid. (c) Use calculus to find the value of b that minimizes the volume of the solid, and compare the result with the answer to part (b). 57. Think About t Match each integral with the solid whose volume it represents, and give the dimensions of each solid. (a) Right circular cylinder m (c) Sphere (e) Torus ' h (i) (ii) 7T dx r2dx ii) 771 (Vr 2 (in (iv) TT (v) TT y=b (b) Ellipsoid (d) Right circular cone x2fdx A [{R + V r 2 - x2)2 -{R- Jr2-x2f]dx 58. Find the volume of concrete in a ramp that is 3 meters wide and whose cross sections are right triangles with base 10 meters and height 2 meters (see figure). Figure for 54 Figure for 56 S 55. Water Depth in a Tank A tank on a water tower is a sphere of radius 50 feet. Determine the depths of the water when the tank is filled to one-fourth and three-fourths of its total capacity. (Note: Use the root-finding capabilities of a graphing utility after evaluating the definite integral.) 59. Find the volume of the solid whose base is bounded by the graphs of y = x + 1 and y = x2-1, with the indicated cross sections taken perpendicular to the x-axis. (a) Squares l r (b) Rectangles of height 1

4 SECTON 6.2 Volume: The Disk Method 431 ^^ 60. Find the volume of the solid whose base is bounded by the circle x2 + y2 = 4, with the indicated cross sections taken perpendicular to the x-axis. (b) Equilateral triangles (a) Squares 64. A manufacturer drills a hole through the center of a metal sphere of radius R. The hole has a radius r. Find the volume of the resulting ring. 65. For the metal sphere in Exercise 64, let R = 5. What value of r will produce a ring whose volume is exactly half the volume of the sphere? 66. The solid shown in the figure has cross sections bounded by the graph of \x\ + \y\ = 1 where 1 < a < 2. 2 (c) Semicircles r y (d) sosceles right triangles 61. The base of a solid is bounded by v = x3, y 0, and x 1. Find the volume of the solid for each of the following cross sections (taken perpendicular to the y-axis): (a) squares, (b) semicircles, (c) equilateral triangles, and (d) semiellipses whose heights are twice the lengths of their bases. 62. Find the volume of the solid of intersection (the solid common to both) of the two right circular cylinders of radius r whose axes meet at right angles (see figure). Two intersecting cylinders Solid of intersection FOR FURTHER NFORMATON For more information on this problem, see the article "Estimating the Volumes of Solid Figures with Curved Surfaces" by Donald Cohen in Mathematics Teacher. To view this article, go to the website Cavalieri's Theorem Prove that if two solids have equal altitudes and all plane sections parallel to their bases and at equal distances from their bases have equal areas, then the solids have the same volume (see figure). c :m;: R2*r H,- " P Area of/?, = area of R2 (a) Describe the cross section when a = 1 and a = 2. (b) Describe a procedure for approximating the volume of the solid. 67. Two planes cut a right circular cylinder to form a wedge. One plane is perpendicular to the axis of the cylinder and the second makes an angle of 6 degrees with the first (see figure). (a) Find the volume of the wedge if (b) Find the volume of the wedge for an arbitrary angle 6. Assuming that the cylinder has sufficient length, how does the volume of the wedge change as 6 increases from 0 to 90?

5 iff f**- SECTON 6.2 VOLUMES Exercises 1-18 mi Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified fine. Sketch the region, the solid, and a typical disk or washer. 1. y = x 2, x = 1, y = 0; about the x-axis 2. x + 2y = 2, x 0 y = 0; about the x-axis 3. y = l/x, x 1, x = 2, y = 0; about the x-axis 4. y = vjc T, x = 2, x = 5, y = 0; aboutthe-x-axis 5. y = * 2, 0 * x =s 2, y = 4, x 0; about the y-axis 6. x = y - y 2, x = 0; about the y-axis 7. y = x 2, y 2 = x; about the x-axis ' 8. y = sec x, y = 1, x = 1, x = 1; about the x-axis 9. y 2 = x, x = 2y; about the y-axis 0. y = x 2/ \ x = 1, y = 0; about the y-axis 1. y = x, y = V*; about y = 1 = x 2, y = 4; about y = 4 - x 4, y = 1; about y = 2 4. y = l/x 2, y = 0, x = 1, x = 3; about y = x = y 2, x = 1; about x = 1 6. y = x, y = Vx; about x = 2 ' y = x 2, x = y 2 ; about x y = x, y = 0, x = 2, x = 4; about x = t 3 about OA 29. ^ 3 about AS 28. & 3 about OC 30. Sft 3 about SC nil Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. 31. y = tan 3 x, y = 1, x = 0; about y = y = (x - 2) 4, 8x - y = 16; about x = y = 0, y = sin x, 0 «x «s ir, about y = y.= 0, y = sinx, 0 ss x =s T; about y = x 2 - y 2 = 1, x = 3; about x = x + 3y = 6, (y - ) 2 = 4 - x; about x = nil Use a graph to find approximate x-coordinates of the points of intersection of the given curves. Then find (approximately) the volume of the solifl obtained by rotating about the x-axis the region bounded by these curves. 37. y = x 2, y = y/x y = x 4, y = 3x - x 3 CASl llll Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line. 39. y = sin 2 x, y = 0, 0 = x =s T; about y = y = x 2-2x, y = x COS(TTX/4); about y = 2 H mi Refer to the figure and find the volume generated by rotating the given region about the specified line llll Each integral represents the volume of a solid. Describe the solid t cos 2 xdx 43. 7rP(y 4 -y % )dy 44. irj^kl + cosx) 2 - l 2 ]<zx 42. 'ify*' 9 - % about OA '^^ bout AB '. about OA S - % about AB 20. % about OC 22. &, about BC 24. 9t 2 about OC 26. a 2 about SC 45. A CAT scan produces equally spaced cross-sectional views of a human organ that provide information about the organ otherwise obtained only by surgery. Suppose that a CAT scan of a human liver shows cross-sections spaced 1.5 cm apart. The liver is 15 cm long and the cross-sectional areas, in square centimeters, are 6, 18, 58, 79, 94, 106, 117, 128, 63, 39, and 0. Use the Midpoint Rule to estimate the volume of the liver. 46. A log 10 m long is cut at 1-meter intervals and its crosssectional areas A (at a distance x from the end of the log) are