Solutions9 - Math 601 Solutions to Homework 9 1 Let R be...

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Math 601 Solutions to Homework 9 1. Let R be the tetrahedron in R 3 with vertices (0 , 0 , 1), (0 , 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 , 0 , 1) , (1 , 1 , 0), and (1 , - 1 , 0), has equation: x + z = 1 The bottom plane, containing the points (0 , 0 , - 1) , (1 , 1 , 0), and (1 , - 1 , 0), has equation: x - z = 1 The side plane that contains the points (0 , 0 , 1) , (0 , 0 , - 1), and (1 , 1 , 0) has equation: x - y = 0 And the other side plane, containing the points (0 , 0 , 1) , (0 , 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 xy -plane. 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). The third side of the triangle is the line x = 1. Thus: Z Z Z R xy 2 dV = Z 1 0 Z x - x Z 1 - x x - 1 xy 2 dz dy dx = 2 45
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2. Let R be the region in R 3
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