HW4 - Math 191 6.2: 4 For the sketch given, c = 0, d = 2 3...

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Math 191 HW 4 Solutions 6.2: 4 For the sketch given, c = 0, d = 3; V = R d c 2 π (shell radius)(shell height) dy = R 3 0 2 πy · ( y 2 ) dy = 2 π R 3 0 y 3 dy = 2 π h y 4 4 i 3 0 = 9 π/ 2. 6.2: 10 a = 0, b = 1; V = R b a 2 π (shell radius)(shell height) dx = R 1 0 2 πx [(2 - x 2 ) - x 2 ] dx = 2 π R 1 0 x (2 - 2 x 2 ) dx = 4 π R 1 0 ( x - x 3 ) dx = 4 π h x 2 2 - x 4 4 i 1 0 = 4 π (1 / 2 - 1 / 4) = π . 6.2: 22 c = 0, d = 1; V = R d c 2 π (shell radius)(shell height) dy = R 1 0 2 πy [(2 - y ) - y 2 ] dy = 2 π R 1 0 (2 y - y 2 - y 3 ) dy = 2 π h y 2 - y 3 3 - y 4 4 i 1 0 = 2 π (1 - 1 / 3 - 1 / 4) = π 6 (12 - 4 - 3) = 5 π/ 6. 6.2: 24 a. V = R d c 2 π (shell radius)(shell height) dy = R 2 0 2 πy h y 2 2 - ± y 4 4 - y 2 2 ²i dy = R 2 0 2 πy ± y 2 - y 4 4 ² dy = R 2 0 2 π ± y 3 - y 5 4 ² dy = 2 π h y 4 4 - y 6 24 i 2 0 = 2 π ± 2 4 4 - 2 6 24 ² = 32 π (1 / 4 - 4 / 24) = 8 π/ 3. b. V = R d c 2 π (shell radius)(shell height) dy = R 2 0 2 π (2 - y ) h y 2 2 - ± y 4 4 - y 2 2 ²i dy = R 2 0 2 π (2 - y ) ± y 2 - y 4 4 ² dy = R 2 0 2 π ± 2 y 2 - y 4 2 - y 3 + y 5 4 ² dy = 2 π h 2 y 3 3 - y 5 10 - y 4 4 + y 6 24 i 2 0 = 2 π (16 / 3 - 32 / 10 - 16 / 4 + 64 / 24) = 8 π/ 5.
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HW4 - Math 191 6.2: 4 For the sketch given, c = 0, d = 2 3...

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