# 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 . Radius r = 2-y The intersection point is x = y 2 ; x=4 y = √x y = √4 WA 1, p. 15 x =4 x = y 2 y=2 rotation about y=2
y = 2 (2 * )* * (2 ( ) ( ) b a b a V radius height thickness V y f y dy 2 2 0 2 2 3 0 2 2 2 3 4 4 3 2 0 0 4 3 2 2 (2 )(4 ) 2 (8 2 4 ) 4 2 2 4 2 8 2 8 2 3 4 4 3 2 (2) 2(2) 4(2) (16) 2(8) 4(4) 2 8(2) 2 16 4 3 2 4 3 2 V y y dy y y y dy y y y y y y y y 16 16 36 16 2 4 8 16 0 2 12 2 3 3 12 12 20 2 3 40 3 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. The intersection point is 2 – x 2 = x x 2 + x -2 =0 (x + 2)(x – 1) = 0 x = -2 , x = 1 WA 1, p. 16
(a) Revolved about the x-axis 2 1 1 2 2 2 2 4 2 0 0 1 1 5 3 2 4 0 0 5 3 2 4 4 5 4 5 4 5 3 (1) 5(1) 1 5 4(1) 4 5 3 5 3 5 25 60 40 15 15 15 15 8 3 b o i a V r dx x x dx x x x dx x x x x dx x (b) Revolved about the y-axis 1 4 3 2 4 3 2 0 2 (1) (1) 2(1) 2 2 4 3 2 4 3 2 1 1 2 3 4 12 12 7 2 2 2 4 3 2 12 12 12 12 12 5 6 x x x V WA 1, p. 17 y = x y = 2-x 2 y = 0 y = -1 x = -1 x = 0
(c) Revolved about the x = -1 1 2 0 1 2 0 1 1 3 2 2 3 2 0 0 1 4 3 2 4 3 2 0 (2 * )* * 2 ( 1)(2 ) 2 ( 1)(2 ) 2 (2 2 ) ( 2 2) 2 (1) 2(1) (1) 2 2 2 2(1) 4 3 2 4 3 2 1 2 1 2 2 4 3 2 b a V radius height thickness V x x x dx x x x dx x x x x x dx x x x dx x x x x 3 8 6 24 2 12 12 12 12 19 38 2 12 12 19 6 V (d) Revolved about the y = -1 2 1 1 2 2 2 2 2 4 2 2 2 0 0 1 1 2 4 2 2 4 0 0 1 5 3 2 5 3 2 0 2 1 (