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
This is the end of the preview. Sign up
access the rest of the document.
This homework help was uploaded on 01/21/2008 for the course MATH 293 taught by Professor Burns during the Fall '08 term at Cornell University (Engineering School).