# Thread: Volume of a sphere with a cyclinder drilled through it

1. ## Volume of a sphere with a cyclinder drilled through it

A ball of radius 13 has a round hole of radius 6 drilled through its center. Find the volume of the resulting solid.

This doesn't seem to be complicated, but as usual, my answers aren't correct.

Volume of the ball is (4/3)*pi*(13^3).

But what's the volume of the hole? Is the hole cylinder or spherical?

2. Originally Posted by thedoge
A ball of radius 13 has a round hole of radius 6 drilled through its center. Find the volume of the resulting solid.
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 $y^2 + x^2 = 13^2$
To me this is a slice of the ball through the center.

Now consider $y = \int_0^6\sqrt{13^2 - x^2}\,dx$
If you rotate this around the y-axis you'll get the volume of the drilled hole.
So the searched volume $V$ will be
$V = \frac{4\pi}{3}13^3 - \pi\int_0^6(13^2 - x^2)\,dx$