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:
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.