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.