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TEXAS A&M UNIVERSITY DEPARTMENT OF MATHEMATICS MATH 251-510/511 Exam 3 version A, Nov 2007 Name: Points: /60 In all questions, no analytical work — no points. 1. For the integral Z 0 8 Z 2 4 ex dxdy, y 1/3 (a) sketch the domain of integration, (b) exchange the order of integration and (c) evaluate the integral. (15 points) 2. Find the area of the part of the plane x + 3y + 2z = 6 in the first octant. (15 points) 3. Find the center of mass of the finite piece of 1-sheet hyperboloid x2 + y 2 − z 2 = 9 cut by the planes z = −1 and z = 2. NB: This is not a type I region so choose your coordinates and order of integration wisely. Due to symmetry, you do not have to calculate x̄ and ȳ. (15 points) 4. Evaluate by performing an appropriate linear change of variables: ZZ (2x2 + xy − y 2 )dxdy, E where the region E is bounded by the lines y = 2x, y = 2x − 1, y = −x and y = 2 − x. (15 points) 5. Bonus question +10%: Derive a formula for the volume of ellipsoid x2 y 2 z 2 + 2 + 2 = 1. a2 b c