Let r be the radius of the circle, and let n be the number of approximating rectangles.

The height h of each rectangle can be defined as:

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

The length $l_k$ of the kth rectangle located at height $y_k$ can be found from

${\left(\frac{l_k^{}}{2}\right)}^2\=r^2-y_k^2$

The area of the kth rectangle is:

$$\begin{gathered} A_{k }\= l_k^{}\cdot h \\ \= 2\cdot \sqrt{r^2-y_k^2} \cdot h \end{gathered}$$
 

Therefore the total area of n rectangles is:

$\mathrm{Area} \= 2 \sum _{k\=1}^n\sqrt{r^2-y_k^2}\cdot h$

In the limit as $n \to \infty$, we get the area of the circle:

$A_{\mathrm{circle}} \=2 {\int }_{-r}^r\sqrt{r^2-y^2} \mathrm{dy}$

$\= \mathrm{\π}\cdot r^2$