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Unformatted text preview: Math 601 Solutions to Homework 9 1. Let R be the tetrahedron in R 3 with vertices (0 , , 1), (0 , , 1), (1 , 1 , 0), and (1 , 1 , 0). Evaluate the following integral: Z Z Z R xy 2 dV Answer: First, it helps to determine the 4 planes that determine the sides of the tetrahedron. The top plane, containing the points (0 , , 1) , (1 , 1 , 0), and (1 , 1 , 0), has equation: x + z = 1 The bottom plane, containing the points (0 , , 1) , (1 , 1 , 0), and (1 , 1 , 0), has equation: x z = 1 The side plane that contains the points (0 , , 1) , (0 , , 1), and (1 , 1 , 0) has equation: x y = 0 And the other side plane, containing the points (0 , , 1) , (0 , , 1), and (1 , 1 , 0), has equation: x + y = 0 Note that each of the above equations only depend on two variables. This makes it relatively easy to set up the integral. For example, if we integrate with respect to z first, then the bounds for z go from the bottom plane to the top place: z = x 1 to z = 1 x . Then, we just need to consider the triangle in the xyplane. Two sides of this triangle are given by the lines x y = 0 and x + y = 0 (these come from the equations for the side planes of the tetrahedron). Thecome from the equations for the side planes of the tetrahedron)....
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 Spring '08
 alndy
 Math, 1 m, dr dθ, Parametric equation, dr dz dθ

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