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TEXAS A&M UNIVERSITY DEPARTMENT OF MATHEMATICS MATH 251-509 Exam 3 version A, 11 Nov 2008 On my honor, as an Aggie, I have neither given nor received unauthorized aid on this work. Name (print): In all questions, no analytical work — no points. 1. For the integral Z 1 1/2 Z 2 yexy dydx, 1/x (a) sketch the domain of integration, (b) exchange the order of integration and (c) evaluate the integral. 2. Find the volume of the body inside both the cylinder x2 + y 2 = 9 and the ellipsoid 4x2 + 4y 2 + z 2 = 100. 3. Evaluate the integral ZZZ (x2 + y 2)dV, H where H is the hemispherical region that lies above the xy-plane and below the sphere x2 + y 2 + z 2 = 1. 4. Evaluate the integral ZZ (y 2 − x2 )dA, D where D is the region between the lines y = x, y = x − 2, y = −x and y = 2 − x. (Hint: while it can be done by other means too, changing the variables makes the integral very easy). 5. Bonus question +10%: Find z = 0, y + z = 1 and x + z = 1. RRR E zdV , where E is bounded by the planes x = 0, y = 0, Points: /20