Please #3, 9, 13, 15, 17, 25, 29, 39, 49, 61

3-4 A solid is obtained by rotating the shaded region about the specied line. (a) Set up an integral using the method of cylindrical shells for the volume ofthe solid. (b) Evaluate the integral to nd the volume of the solid. 3. About the yaxis 4. About the xaxis ). y = costxzt 5-8 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 specied line. 5. v = lnx. v = O. x = 2'. about the vaxis 6. v = .r'l, v = 8. x = 0; about the xaxis 7. y = Sinll. y = 77/2. x = 0; about y = 3 8. y=4x .r\". y =x; aboutx= 7 9-14 Use the method of cylindrical shells to nd the volume generated by rotating the region bounded by the given curves about the raids. 9. y=, _v=0, x=4 10. y =x". y: 0. .1: =1. .t = 2 11. )'= l/x, y = 0, x =1, x=4 12. y = ('2. y = O. x = O. .r =1 13. y=\\/W. y=0. .r=0. x=2 14. y = 4x x1. _v = .1: 15-20 Use the method of cylindrical shells to nd the volume of the solid obtained by rotating the region bounded by the given curves about the .r-axis. 15. ne=1, _r=0, y=l. y=3 16. y=\\/.?, .r=0. v=2 17. y=xm, _v=8, _r=0 18. X: 3_v2 +12y 9. .r = 0 19. _t=l+(_v2)l, x=2 20. x+y=4. x=__vl4_v+4 21-22 The region bounded by the given curves is rotated about the specied axis. Find the volume ofthe resulting solid using (a) x as the variable of integration and (b) y as the variable of integration. 21. v = xi, v = SJ}: about the v-axis 22. v =15, \\' = 4x"; about the x~axis 2324 A solid is obtained by rotating the shaded region about the specied axis. (a) Sketch the solid and a typical approximating cylindrical shell. (b) Use the method of cylindrical shells to set up an integral for the volume of the solid. (c) Evaluate the integral to nd the volume. 23. About x\"\" 2 24. About v = l y y = 4x x: 2530 Use the method of cylindrical shells to nd the volume generated by rotating the region bounded by the given curves about the specied axis. 25. v =1 'q ill 37. Use the Midpoint Rule with H : 5 to estimate the volume obtained by rotating about the v-axis the region under the curvey= J1 + x-KOS .v S l. 38. If the region shown in the gure is rotated about the r-axis to form a solid. use the Midpoint Rule with r: = 5 to esti- mate the volume of the solid. 3942 Each integral represents the volume of a solid. Describe the solid. 39. l: 2am:s 11" 40. f: 211'}! In); dy o 4 *+ 2 41. 2w|ll _, d)- . v- 42. [0' 21112 my 2") a: . 43-44 Use a graph to estimate the .t-coordinates of the points of intersection of the given curves. Then use this information and a calculator or computer to estimate the volume of the solid obtained by rotating about the )'-axis the region enclosed by these curves. 43. y = .r2 2.1: y = T x+l 44. y : 8"\". y :Jr\"I - 4.1: + 5 45-46 Use a computer algebra system to nd the exact volume of the solid obtained by rotating the region bounded by the given curves about the specied line. 45. y = sinlx. y = sin\".r. O S x 5. 11'; about x = 17/2 46. )9 =.r"sin.r. y= 0. 0 E IE 1:". aboutx= 1 47-52 A solid is obtained by rotating the shaded region about the specied line. (a) Set up an integral using any method to nd the volume of the solid. (b) Evaluate the integral to nd the volume of the solid. 47. About the y-axis 48. About the .r-axis 51. About the line .I = 2 52. About the line y = 3 .l. 53 -59 The region bounded by the given curves is rotated about the specied axis. Find the volume ofthe resulting solid by any method. 53. v = ix" + 6x 7 8. v = 0: about the v-axis 54. v = ix" + 6x 7 8. v = 0; about the .r-axis F r 55. v i x' = l, v = 2: about the .r-axis 56. \\"F v x" = l, v = 2: about the vaxis 57. Jr" + (v l): = 1; about the v-axis 58. x = (v 3):. .r = 4; about 3' = l 59. x=(v1)I. .ry= 1; aboutx=l 60. Let T be the triangular region with vertices (0, O). (l. 0), and ( l, 2). and let V be the volume of the solid generated when T is rotated about the line it = a. where a > 1. Express 0 in terms of V. 61 63 Use cylindrical shells to nd the volume of the solid. 61. A sphere of radius r