# Having trouble with this multiple integral question

## 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.

## Answers and Replies

Office_Shredder
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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

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.