6

The function $f$ is defined as $f(x)=\sqrt{x\sin\bigl(x^{2}\bigr)}$, where $0\le x\le\sqrt{\pi}$.
Consider the shaded region $R$ enclosed by the graph of $f$, the $x$-axis and the line $x=\frac{\sqrt{\pi}}{2}$, as shown below.

The shaded region $R$ is rotated through $2\pi$ radians about the $x$-axis to form a solid.

Show that the volume of the solid is $\displaystyle \frac{\pi\,(2-\sqrt{2})}{4}$.