tut10 - D , if the density is ρ ( x, y, z ) = x 2 + y 2 +...

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MATH 237 Tutorial 10 Wednesday, July 19th, 2006 1) Evaluate Z Z R Z 1 ( x + y + z ) 3 dV where R is the region bounded by the six planes z = 1 , z = 2 , y = 0 , y = z, x = 0 and x = y + z . 2) Evaluate the following integral by changing the order of integration: Z 1 0 Z 1 - x 0 Z 1 y sin( πz ) z (2 - z ) dzdydx . 3) Let D be the region inside the cylinder x 2 + y 2 = 1 and between the planes z = 1 and z = y + 2. a) Set up in Cartesian coordinates the double integral that computes the volume of D . b) Set up in Cartesian coordinates the triple integral that computes the mass of
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Unformatted text preview: D , if the density is ρ ( x, y, z ) = x 2 + y 2 + 1. c) Use cylindrical coordinates to evaluate the integral in (b). 4) Let D be the region in R 3 which lies inside the sphere x 2 + y 2 + z 2 = 2 b 2 and outside the sphere x 2 + y 2 + z 2 = b 2 ( b > 0) with x and y restricted to x ≥ , y ≥ 0. Evaluate Z Z D Z 1 x 2 + y 2 + z 2 dV . 5) Evaluate Z Z D Z e ( x 2 + y 2 + z 2 ) 3 / 2 dV where D is the region x 2 + y 2 + z 2 ≤ 1....
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This note was uploaded on 05/19/2011 for the course MATH 237 taught by Professor Wolczuk during the Spring '08 term at Waterloo.

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