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Unformatted text preview: Fall 2010 Math 251 Exam 3A: Solutions Tue, 09/Nov c 2010 Art Belmonte Refer to diagrams drawn on your exam. 1. Find the volume of the solid in the first octant bounded by the circular cylinders x 2 + z 2 = 9 and y 2 + z 2 = 9, and the coordinate planes. Rectangular coordinates are recommended. Use the quarter-circular region in the yz-plane as your shadow region. (Alternatively, use a similar region in the xz-plane or a square in the xy-plane.) The volume is V = ZZZ E 1 dV = Z 3 Z 9- y 2 Z 9- z 2 1 dxdzdy = 18 cm 3 . 2. Evaluate the integral Z ln10 Z 10 e x 1 ln y dydx by reversing the order of integration. Express the curve y = e x as x = ln y . Then Z 10 1 Z ln y 1 ln y dxdy = 9. 3. Determine the mass and center of mass of the finite region in the xy-plane bounded by the curves y = 1 1 + x 2 and y = 1 2 x 2 if its density is = y . When the curves intersect, their y-coordinates are equal. Thus 1 1 + x 2 = 1 2 x 2 , whence x = 1....
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