In here, question 6, they calculate the volume of a torus by using cross-sections across the x-axis.
However, I have found an easier way to do it, using the same diagram:
$$\int_0^{2\pi R} \pi r^2 \, dR = 2\pi^2 Rr^2.$$
(I skipped a few steps because I couldn't use begin{align} and end{align}.)
Why did they do it calculating cross-sections instead of using this easier method?
They wanted to use a more rigorous method, as using pure geometry will not always work.