Since [tex]\sin^2(\sin x)[/tex] 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
[tex]y\approx\begin{cases}\dfrac{2\sin1}\pi x&\text{for }0\le x<\dfrac\pi2\\\\2\sin1\left(1-\dfrac1\pi x\right)&\text{for }\dfrac\pi2\le x<\pi\end{cases}[/tex]
Then the volume is approximately
[tex]\pi\left(\int_0^{\pi/2}\left(\dfrac{2\sin1}\pi x\right)^2\,\mathrm dx+\int_{\pi/2}^\pi\left(2\sin1\left(1-\dfrac1\pi x\right)\right)^2\,\mathrm dx\right)[/tex]
which evaluates to [tex]\dfrac{\pi^2\sin^21}3\approx2.329[/tex]. The actual volume is slightly larger, since [tex]\sin^2(\sin x)[/tex] 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 [tex]\sin^2(\sin x)[/tex] with, say, a 4th degree polynomial that would be easy to integrate and yield a result much closer to the actual value.
