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**Jhevon** $\displaystyle r^2 + z^2 = a^2$ is simply a sphere of radius $\displaystyle a$ centered at the origin.

following **Mr F's** lead, the cylinder is $\displaystyle x^2 + \left( y - \frac a2 \right)^2 = \left( \frac a2 \right)^2$

in the xy-plane, the trace of this cylinder is a circle with radius a/2 centered at (0, a/2)

now, you are bounded above and below by the sphere. we can simply find the volume in the first quadrant, and then multiply the answer by 4.

hence, we integrate the function 1, with the following limits:

$\displaystyle 0 \le x \le \sqrt{ay - y^2}$, $\displaystyle 0 \le y \le a$ and $\displaystyle 0 \le z \le \sqrt{a^2 - x^2 - y^2}$

and don't forget to multiply by 4

any questions?