Set up the integral ∫∫_S (yz)dS if S is the part of the paraboloid x^2 + y^2 = z with 0 <= z <= 1.

Question

Let S be the part of the paraboloid x^2 + y^2 = z with 0 <= z <= 1. Set up but do not evaluate the integral ∫∫_S (yz)dS.

Answer 1

You can guess geometrically that the answer should be zero, since the average y value on a surface of revolution around the z-axis is zero. Basically, at each level set, the integral of the function is zero, so "adding these up" doesn't change the value. If you want to be formal, in cylindrical coordinates, we have t he map

(theta,r) --->(theta,r,r^2). For the adjustment, you should take the norm of the cross product of partial derivatives, giving you dS=sqrt(1+4r^2)dA, and dA=r dr dtheta. Putting everything together gives you the integral

int_0^1 int_0^{2pi} r^4sin(theta)*sqrt(1+4r^2)dtheta dr

evaluating sintheta first from 0 to 2pi will give you zero, and integrating again doesn't change that