Volumes of Regions with Known Cross Sections

Find the volume of the solid bounded by the curves y = x^2 and y = 8 - x^2 who cross sections are perpindicular to the x - axis with one side on the xy plane using semi circle cross sections

Ok so i have to find total volume which is equal to the Rheiman sum of the the volume of the semi circle which is

$\displaystyle \frac{1}{2}\pi*r^2*\delta thickness$

and where delta thickness is the change in x?

and the radius of the semi circle is 1/2 the height between y = 8 - x^2 and y = x^2

which gives

$\displaystyle \frac{8-2x^2}{2}$

$\displaystyle 4-x^2$

then you will have the intergral from x = -2 to x = 2 because this is where they intersect

$\displaystyle \frac{\pi}{2}\int{4-x^2}^2$

For some reason I dont think im accurately doing this