Integration with square root involved

Question

How to compute the integral of $\sqrt{1-r^2}$ with respect to $r$? Is there a substitution? What are the steps? Thanks

Answer 1

The standard substitution is

$$ r = \sin t \implies dr = \cos t dt$$

Hence

$$ \int \sqrt{1-r^2} = \int \sqrt{1 - \sin^2 t } \cos t dt = \int \cos^2 t dt$$

and this is an easier integral since you can use the fact that

$$ \cos^2 t = \frac{1 - \cos 2t}{2} $$