Calculus 3 (Math-UA-123.005) Spring 2017 Homework 8: Sections 12.4-12.6 Give complete, well-written solutions to the following exercises. The homework has to be handed in on April 6 at the beginning of the class. 1. Evaluate the following triple integral. Note that you can evaluate the result by using only geometric interpretation and symmetry, without explicitly performing the integration. ZZZ (4 + 5x2 yz 2 ) dV, C 2 where C is the cylindrical region: x + y 2 4, 2 z 2. 2. Find the center of mass ( x, y, z) of the solid body C bounded between the planes z = 1 + x + y and x + y + z = 1, and above the triangle x + y 1, x 0, y 0 in the xy-plane, assuming it has unit mass RRR density. Remember that the x-component x(x, y, z)RRR dV and similarly for the other of the center of mass is given by x = M1 C components y and z, where M is the total mass, M = (x, y, z) dV . C 3. Convert Z 1 Z 1y2 Z x2 +y2 xyz dz dx dy 1 0 x2 +y 2 into an integral in cylindrical coordinates. 4. What is the total charge of a body E with charge density (x, y, z) = y, where E is the region that lies between the plane z = x + 2 and the xy-plane, and between the cylinders x2 + y 2 = 1 and x2 + y 2 = 4. 5. RRR Set up the iterated integral for evaluating f (r, , z) dW over the region D, deD scribed as follows: D is the solid cylinder whose base is the region in the xy-plane between the circles r = cos and r = 2 cos and whose top lies in the plane z = 3 y. p 6. A solid is bounded from below by the cone z = x2 + y 2 and from above by the plane z = 1. The density of the solid is given by (r, , z) = z 2 . Find the total mass M and the average density (total mass per volume) of the solid. The homework has to be handed in on April 6 at the beginning of the class. 1