# How do you find the area of circle using integrals in calculus?

##### 1 Answer
Sep 8, 2014

By using polar coordinates, the area of a circle centered at the origin with radius $R$ can be expressed:
$A = {\int}_{0}^{2 \pi} {\int}_{0}^{R} r \mathrm{dr} d \theta = \pi {R}^{2}$

Let us evaluate the integral,
$A = {\int}_{0}^{2 \pi} {\int}_{0}^{R} r \mathrm{dr} d \theta$
by evaluating the inner integral,
$= {\int}_{0}^{2 \pi} {\left[\frac{{r}^{2}}{2}\right]}_{0}^{R} d \theta = {\int}_{0}^{2 \pi} {R}^{2} / 2 d \theta$
by kicking the constant ${R}^{2} / 2$ out of the integral,
${R}^{2} / 2 {\int}_{0}^{2 \pi} d \theta = {R}^{2} / 2 {\left[\theta\right]}_{0}^{2 \pi} = {R}^{2} / 2 \cdot 2 \pi = \pi {R}^{2}$