Question

How do you find the area of a circle using integration?

Answer 1

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`