Unformatted text preview: 302 Section 7.3
19. 15. A cross section has radius r A(y)
1 0 tan 4 y and area r2 tan2
4 tan2 y dy 4 4 4 y . The volume is
1 tan 1 4 y y
0 [ 6, 6] by [ 4, 4] The solid is a sphere of radius r 4 . 20.
4 3 r 3 3. The volume is 36 . 16. A cross section has radius r A(x) from x x
2 sin x cos x and area r2 sin2 x cos2 x. The shaded region extends 0 to where sin x cos x drops back to 0, i.e., where
[ 0.5, 1.5] by [ 0.5, 0.5] 2 . Now, since cos 2x
1 2 cos2 x 1, we know The parabola crosses the line y x x
2 0 when 0 or x 1. A cross cos 2x cos x and since cos 2x 1 2 sin2 x, we 2 /2 1 cos 2x know sin2 x . sin2 x cos2 x dx 0 2 /2 1 cos 2x 1 cos 2x dx 0 2 2
/2 x(1 x) 0, i.e., when x x x 2)2 x and area (x 2
1 3 x 3
2 section has radius r A(x)
1 0 r2 (x 2x 3
1 4 x 2 x 4).
1 5 x 5
1 0 4 4 8 0 (1
/2 cos2 2x) dx
cos 4x dx 2
/2 0 /2 The volume is (x 2 2x 3 x 4) dx
30 4
/2 0 sin2 2x dx cos 4x) dx
2 . 1 0 8 8 0 (1 0 21. x 1 sin 4x 4 2 0 16 . 17. [ 1, 2] by [ 1, 2] Use cylindrical shells: A shell has radius y and height y. The volume is
1 [ 2, 4] by [ 1, 5]
0 2 (y)( y) dy x 2 and area 22. 2 1 3 y 3 1 0 2 . 3 A cross section has radius r A(x)
2 0 r2 x4. The volume is
1 5 x 5
2 0 x 4 dx 32 . 5
[ 1, 3] by [ 1, 3] 18. Use washer cross sections: A washer has inner radius r outer radius R
1 x, 2x, and area A(x) 3 x 2 dx 3
0 (R 2
1 r 2) . 3 x 2. [ 4, 6] by [ 1, 9] The volume is x 3 and area 1 3 x 3 0 A cross section has radius r A(x)
2 0 r2 x 6. The volume is
1 7 x 7
2 0 x 6 dx 128 . 7 ...
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 Spring '08
 GERMAN
 Cos, dx

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