PS9_answers - A4. 100. At all points on the sphere x 2 + y...

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Math 237 Problem Set 9 Answers A1. i) R 3 - 1 R 3 - 3 R 0 - 2 f ( x, y, z ) dz dy dx . ii) R b - b R a q 1 - y 2 b 2 - a q 1 - y 2 b 2 R 3 1 f ( x, y, z ) dz dx dy. iii) R b 0 R a (1 - y b ) 0 R c ( 1 - x a - y b ) - c ( 1 - x a - y b ) f ( x, y, z ) dz dx dy. iv) R 5 0 R tan - 1 (1 / 2) 0 R 2 π 0 ρ 2 sin φf ( ρ sin φ cos θ, ρ sin φ sin θ, r cos φ ) dθ dφ dρ v) R 2 π 0 R 2 0 R 1 - r 2 - 3 - r 2 rf ( r cos θ, y, r sin θ ) dy dr dθ. A2. R 1 0 R 1 - x 0 R 1 - x - y 0 e x dz dy dz and R 1 0 R 1 - y 0 R 1 - y - z 0 e x dx dx dy = e - 5 2 . A3. The region of integration is the region bounded by the cone x 2 - (1 - y ) 2 + z 2 = 0 the plane y = 0. In the order dydxdz we get Z - 1 1 Z 1 - z 2 - 1 - z 2 Z 1 - x 2 + z 2 0 f ( x, y, z ) dy dx dz.
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Unformatted text preview: A4. 100. At all points on the sphere x 2 + y 2 + z 2 = 1. A5. 5 π 192 A6. 4 π (ln( a/b )). A7. i) π 3 ( √ 2-1) ii) π 4 √ 3 . A8. π 3 h 3 . B1. iii) 1 2 πk . B4. 4 3 πabc B6. 4 5 e 2 ( e-1) B8. b) b ± 1 ln 2-1 ² 1 / 3 . B9. ( π 6 b 3 + 15 32 b 5 ) πρ . 1...
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This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.

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