Looks like the region in question is the one bounded by [tex]y=5\sqrt{\sin x}[/tex] and the [tex]x[/tex]-axis, between [tex]x=\frac\pi3[/tex] and [tex]x=\frac{3\pi}4[/tex]. Revolving about the [tex]x[/tex]-axis is trivial using the disk method:
[tex]\displaystyle\pi\int_{\pi/3}^{3\pi/4}(5\sqrt{\sin x})^2\,\mathrm dx=25\pi\int_{\pi/3}^{3\pi/4}\sin x\,\mathrm dx=\boxed{\frac{25(1+\sqrt2)\pi}2}[/tex]
