The problem is that my answer is wrong when i try to set up an integral when thinking in terms of cylinderical shells:

We have a rectangle, bent in the shape of a circle, with length [tex]2 \pi r[/tex] height [tex]\sqrt{1-x^2}[/tex] and width[tex] \Delta x[/tex], so it's volume should be all that multiplied together.

I have torus radius (this torus is just a circle, really) of R and the circle being revolved has a radius of r. So my integral is:

[tex]\int_{R-r}^{R+r} 4 \pi x \sqrt{1-x^2} \Delta x[/tex]

and the above is off by [tex]r^3 R[/tex] when i use differences of two volumes, what conceptual flaw am I making?