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

By using polar coordinates, the area of a circle centered at the origin with radius ##R## can be expressed: ##A=int_0^{2pi}int_0^R rdrd theta=piR^2##

Let us evaluate the integral, ##A=int_0^{2pi}int_0^R rdrd theta## by evaluating the inner integral, ##=int_0^{2pi}[{r^2}/2]_0^R d theta=int_0^{2pi}R^2/2 d theta## by kicking the constant ##R^2/2## out of the integral, ##R^2/2int_0^{2pi} d theta=R^2/2[theta]_0^{2pi}=R^2/2 cdot 2pi=piR^2##