Assume the center of the sphere is located at the origin.

Let r be the radius of the sphere, and let n be the number of approximating cylinders.

The height h of each cylinder can be defined as:

$h \= \frac{2\cdot r}{n}$

The radius $r_k$ of the kth cylinder located at height $y_k$ can be found from

$r_k^{2 }\=r^2-y_k^2$

The volume of the kth cylinder is thus:

$$\begin{gathered} V_{k }\= {\mathrm{\pi }}_{}\cdot r_k^2\cdot h \\ \= \mathrm{\π}\cdot \left(r^2-y_k^2\right)\cdot h \end{gathered}$$

Therefore the total volume of n cylinders is:

$\mathrm{Volume} \= \sum _{k\=1}^n{\mathrm{\π}}_{} \cdot \left(r^2-y_k^2\right)\cdot h$

In the limit as $n \to \infty$, we get the volume of the sphere:

$V_{\mathrm{sphere}} \= {\int }_{-r}^r\mathrm{\π} \cdot \left(r^2-y^2\right) \mathrm{dy}$

$\= \frac{4}{3}\cdot \mathrm{\pi }\cdot r^3$