I am supposed to find the volume of a torus with radii r and R using cylindrical shells. I don't even know what a torus is, and the whole shell method is still confusing to me.
The plot is a cross-section of a torus showing a cross-section of one of the cylindrical shells (thick black walls) we'll use to calculate the volume. The equation of the cross-section on the right is:
$\displaystyle (x-R)^2+y^2=r^2$ or $\displaystyle y=\sqrt{r^2+(x-R)^2}$
and the formula for calculating the volume using cylindrical shells is:
$\displaystyle V=2\pi\int_a^b x f(x)dx$
Now, in this case, that would be:
$\displaystyle V=2\left(2\pi\int_{R-r}^{R+r} x\sqrt{r^2-(x-R)^2}dx\right)$
since we're integrating only the top half right?
which is messy and also you get an $\displaystyle \arctan$ component which you have to take the limit of as x goes to either limit. Eventually you get:
$\displaystyle V=4\pi\left(1/4\pi r^2 R-(-1/4\pi r^2 R)\right)=2\pi^2 Rr^2$
Maybe an easier way though.