In a problem regarding volumes of revolution, we are asked to estimate the volume capacity of the bowl shown by taking the average of right- and left-endpoint approximations to the integral with $N=7$. We are given the inner radii starting from the top $0, 4, 7, 8, 10, 13, 14, 20$, and these two graphs,
In the solutions manual, starting with the right-endpoint, $$R_{7}=3\pi \left((23^{2}-14^{2})+(25^{2}-13^{2})+(26^{2}-10^{2})+(27^{2}-8^{2})+(28^{2}-7^{2})+(29^{2}-4^{2})+(30^{2}-0^{2}) \right)=13470\pi$$
I am not asking for the full solution, I want to understand what did we exactly do here? Where did the $3\pi$ come from? And where did we get values such as $23$ and $26$.. etc? What do we mean by $23^{2}-14^{2}$? What does this represent?
I am confused because the textbook did not solve a similar problem where we used average of right and left-endpoint approximations. There is no solved example I can understand from.