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Math 241, Quiz 9. 10/31/12. Name: • Read problems carefully. Show all work. No notes, calculator, or text. • There are 15 points total. 1. §15.3, #22 (7 points): Find the volume of the solid enclosed by the paraboloid z = x2 + 3y 2 and the planes x = 0, y = 1, y = x, and z = 0. Solution: We have 1 Z Z 1 2 Z 2 1 2 (x + 3y ) dy dx = 0 3 1 (x y + y ) x y=x 1 = 3 2 (x + 1) − 2x3 dx dx = 0 Z 1 Z 2 (−2x + x + 1) dx = 0 0 1 x3 − x4 + +x 2 3 1 x=0 5 1 1 =− + +1= . 2 3 6 2. §15.3, #48 (8 points): Evaluate the double integral Z 1Z 1 ex/y dy dx 0 x by reversing the order of integration. Solution: We compute Z 1 Z y x/y e 0 0 Z dx dy = 1 x/y y ye 0 Z dy = x=0 0 1 1 (ye − y) dy = (e − 1)(y /2) 2 y=0 = e−1 . 2