Question

Please help and show steps!

Prove that the area of a circle of radius r is Πr^2 using integration by trig substitution

Answer 1

Consider the semicircle of radius \(r\) described by \(y=\sqrt{r^2-x^2}\). Integrating from \(x=-r\) to \(x=r\) will give the area of the semicircle, so double that to get the area of a full circle.
\[A=2\int_{-r}^r\sqrt{r^2-x^2}~dx\]
The trig sub would be \(x=r\sin u\) or \(x=r\cos u\) (either will work, though the sine sub is more often used).