Proving the shell method using triple integrals

[morsel]

Homework Statement

The volume of a solid of revolution using the shell method is [tex]\int_{a}^{b} 2\pi x f(x) dx[/tex]. Prove that finding volumes by using triple integrals gives the same result. (Use cylindrical coordinates with the roles of y and z changed).

Homework Equations

[tex]dV = r dr d\theta dz[/tex]

The Attempt at a Solution

[tex]x = r cos\theta[/tex]

[tex]y = y[/tex]

[tex]z = r sin\theta[/tex]

I'm not sure if this is changing the roles of y and z... and I don't know how to proceed from here. Any hints?

 

[lippyka]

Edit: Ok, I think I figured it out. \int_{a}^{b} \int_{0}^{2\pi}\int_{0}^{f(r cos\theta)} r dr d\theta dz = \int_{a}^{b} \int_{0}^{2\pi} 2\pi r f(r cos\theta) d\theta dr = \int_{a}^{b} 2\pi r f(x) dx