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:
or
and the formula for calculating the volume using cylindrical shells is:
Now, in this case, that would be:
since we're integrating only the top half right?
which is messy and also you get an component which you have to take the limit of as x goes to either limit. Eventually you get:
Maybe an easier way though.