What is the pi times the definite integral from 0 to pi of sin(sin(x)^2 dx

Since doesn't have an elementary antiderivative, it's unlikely that there is an exact answer here. However, there are some facts you can use to make a good approximate guess.

One thing you can do is approximate the function by



Then the volume is approximately



which evaluates to . The actual volume is slightly larger, since is slightly greater than this approximating function. The closest answer is 3.830. Indeed, the actual value to 20 decimal places is about 3.8299454909568467491, so this is the correct answer.

Note that a better approximating function would yield a better solution. If you know about Taylor series, you can use that to your advantage and approximate with, say, a 4th degree polynomial that would be easy to integrate and yield a result much closer to the actual value.