How to express volume of a sphere as a sum of infinitesimally thick discs?

Question

I want to express the volume of a sphere with a radius r as an integral that adds up each infinitesimally thick disc within the volume. So I have dV = A(x) dx, where A(x) is the area of the disc that is at coordinate x. I'm having trouble finding this function in terms of r. Remember, r is the radius of the sphere. At x=0, A = πr^2, and at x=r, A = 0. The bounds of the integral should be -r to r I believe. But what is A(x)?

Answer 1

\begin{eqnarray*} V=\sum \pi (r^{2}-x^{2}) \delta x=\pi \int_{-r}^{r} (r^{2}-x^{2}) dx \end{eqnarray*} A little integral calculus & you will recover the well known result. $V_S= \frac{4}{3} \pi r^3$.