Math 293 Practice Prelim #1 Spring 2000 Formulas that may or may not be useful: x = ρ sin φ cos θ y = ρ sin φ sin θ z = ρ cos φ dx dy dz = ρ 2 sin φdρdφdθ 1 . Consider the integral Z 20 Z 4-x 20 xe 2 y 4-y dy dx . (a) Sketch the region of integration. (b) Evaluate the integral. 2 . Consider the integral Z 10 Z √ 1-x 20 e-( x 2 + y 2 ) dy dx . (a) Change this integral into an equivalent polar coordinate integral. (b) Evaluate the integral. 3 . Let R be the region in 3 dimensional space bounded below by the xy-plane, bounded above by the cone z = p x 2 + y 2 , and lying between the spheres x 2 + y 2 + z 2 = 9 and x 2 + y 2 + z 2 = 16. Find the moment of inertia I z = ZZZ R ( x 2 + y 2 ) δdxdydz with the density δ =1. 4 . Use Green’s theorem to calculate the counterclockwise circulation of the vector ﬁeld
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