12 use cylindrical shells to compute the volume of

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12. Use cylindrical shells to compute the volume of the region bounded by 2 x y and x = 4 , revolved about y = 2 . WA 1, p. 10 2 1 2 1 1 1 1 23 23 1 1 1 1 1 3 4 1 1 2() 2 , 22 22 22 8 42 343 b a Vx fxd x rxhx Vxxd x Vxxd xVxd xxd x xx V                       2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 , 4 2 4 24 4 4 4 4 44 4 16 4 4 4 16 2 4(0) 32 r yh y V y ydy V y ydy V y y ydy V ydy y ydy V         
2 2 2 2 2 0 0 2 3 2 0 2 4 3 2 0 2 , 4 2 2 4 2 2 4 2 2 4 8 2 16 2 2 8 2 4 8 16 0 4 3 3 36 16 40 2 3 3 3 r y h y V y y dy y y dy V y y y dy y y V y y V 22. Use the best method available to find the volume of the region bounded by 2 2 , ( 0) y x y x x and the y -axis revolved about (a) the x -axis, (b) the y -axis, (c) x = –1, and (d) y = –1. a. washer method WA 1, p. 11
2 2 2 2 1 1 2 2 0 0 1 1 4 2 2 0 0 1 1 5 3 3 0 0 3 5 3 2 , 2 4 4 4 4 5 3 3 4 1 1 1 43 5 4 1 0 0 5 3 3 15 15 38 15 o i b b o i a a r x r x V r dx r dx V x dx x dx V x x dx x dx x x x V x V V b. shell method 2 1 2 0 1 3 2 0 1 4 3 2 0 , 2 2 2 2 2 10 5 2 0 4 3 12 6 r x h x x V x x x dx V x x x dx x x V x c. shell method 2 1 2 0 1 3 2 0 1 3 2 2 4 3 4 0 , 2 2 1 2 2 2 2 2 1 2 1 1 2 2 2 2 1 0 2 4 3 2 4 3 19 6 r x h x x V x x x dx V x x x dx x x x V x V d. washer method WA 1, p. 12
2 2 2 2 1 1 2 2 0 0 1 1 4 2 2 0 0 1 1 5 3 3 2 0 0 5 3 3 2 2 1, 1 3 1 6 9 2 1 2 9 5 3 1 1 2 1 9 1 0 1 1 0 5 3 1 1 36 7 2 5 3 5 o i b b o i a a r x r x V r dx r dx V x dx x dx V x x dx x x dx x x V x x x x V V 7 73 3 15 24. Use the best method available to find the volume of the region bounded by 1, 2 x y e y x and the x-axis revolved about the (a) x -axis and (b) y -axis.

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