Having trouble with this multiple integral question

[russia123]

Homework Statement

Given a point P(x,y,z) in a three-dimensional space, let P' denote the projection of P onto the xy-plane and let O denote the origin of the coordinates, and define f(x,y,z) as the area of the triangle. Compute:

Integral of the Integral of the Integral of f(x,y,z) under E

Where E is the portion of the solid cylinder x^2+y^2=y lying between the horizontal planes z = 0 and z = 1.

2. The attempt at a solution

Okay, now I've gotten the equation of the area of the triangle: z*sqrt(x^2+y^2)/2, and then I realized that I need to change this to polar coordinates, hence:

sqrt(x^2+y^2) = r, under the assumption that r is always positive.

Due to this we get the triple integral of z*r/2*r*dr*dtheta*dz = the triple integral of z*r^2/2*dr*dtheta*dz.

Now, I've gotten to the point where my bounds for dr are 0 to sin(theta), dtheta are 0 to 2pi and z are 0 to 1, but for some reason I come out to 0. This comes from my bounds for theta being 0 to 2pi and I don't understand why this is wrong.

 

[Office_Shredder]

f(x,y,z) is the area of what triangle?

Looking at your last integral, the integral from 0 to 2pi of dtheta is just 2pi, not 0. There are no trigonometric functions or anything that cancels out your integral

 

[russia123]

It is the area of the triangle formed by O, P' and P.

Well, what happens, when I try to solve it is that I need to take the integral of z*sin(theta)^3/6 with respect to theta from 0 to 2pi and that goes to 0 for me.