Unformatted text preview: Chapter 7 Review
17. Solve x 3 x 0 and x
x x2 1 319 x to find the intersection points at (c) Use cylindrical shells. A shell has radius 4 height 2 x
4 x and 21/4. Then use the area's symmetry: x. The total volume is x) dx 2x 3/2
4 5/2 x 5 2 (4 the area is
2 1/4 0 4 x)(2 x (8 x 4x 2x 2 2
0 x 2) dx
1 3 x 3
4 0 2
0 x x2 1 1 ln (x 2 2 (x3 1) x) dx
1 4 x 4 1 2 x 2
21/4 0 2 16 3/2 x 3 64 . 5 2 (d) Use cylindrical shells. A shell has radius 4 height y
y2 . The total volume is 4 4 y2 2 (4 y) y dy 4 0 4 y3 2 4y 2y 2 dy 4 0 4 2 3 32 1 4 2 2y 2 y y . 3 3 16 0 y and ln ( 2 1) 2 1 1.2956. 18. Use the intersect function on a graphing calculator to determine that the curves intersect at x The area is
1.8933 1.8933 1.8933. 31 x2 x2 3 dx, 10 which using NINT evaluates to 5.7312. 22. (a) Use disks. The volume is
2 0 k 19. Use the x and yaxis symmetries of the area: 4
0 ( 2y)2 dy y2 k , 1,
k 0 2 0 2y dy k2 y2 2 4 .
0 x sin x dx 4 sin x x cos x
0 4 . (b) 0 2y dy 20. A cross section has radius r A(x)
1 3x 4 and area (c) Since V When k 2 . 2 dV r
1 2 8 9 x . x
9
1 1 8 dt V 21. 9 x dx dk dt dk dt 1 dV 2 k dt 2 k .
1 (2) 2 1 1 , so the depth is increasing at the rate of second. unit per 23. The football is a solid of revolution about the xaxis. A
[ 5, 5] by [ 5, 5] cross section has radius r2 12 12
0 12 1 The graphs intersect at (0, 0) and (4, 4). 1 1 (a) Use cylindrical shells. A shell has radius y and height y
y . The total volume is 4 4 4 y2 y3 2 (y) y dy 2 y2 dy 4 4 0 0 4 1 3 1 4 2 y y 3 16 0 32 . 3
2
11/2 4x 2 and area 121 2 4x 2 . The volume is, given the symmetry, 121 11/2 4x 2 4x 2 dx 24 1 dx 121 121 0 11/2 2 1 2 24 x x3 3 11 0 11 11 24 2 6 88 276 in3. 24. The width of a cross section is 2 sin x, and the area is (b) Use cylindrical shells. A shell has radius x and height 2 x
4 x. The total volume is
4 2 (x)(2 x
0 x) dx 2
0 (2x 3/2 x ) dx
1 3 x 3
4 0 2 1 2 1 r sin2 x. The volume is 2 2 1 1 1 sin2 x dx x sin 2x 2 2 4 0 2 2 0 4 . 2 4 5/2 x 5 128 . 15 ...
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 Spring '08
 GERMAN
 Trigraph, Prism

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