Originally Posted by

**fgn** It will not be a cylinder. Look at the top/bottom of the thing that you have drilled out. It will not be flat, it will be curved just like the ball.

First look at the circle $\displaystyle y^2 + x^2 = 13^2 $

To me this is a slice of the ball through the center.

Now consider $\displaystyle y = \int_0^6\sqrt{13^2 - x^2}\,dx $

If you rotate this around the y-axis you'll get the volume of the drilled hole.

So the searched volume $\displaystyle V $ will be

$\displaystyle V = \frac{4\pi}{3}13^3 - \pi\int_0^6(13^2 - x^2)\,dx $