Volume of sphere remaining after boring a cylinder through its center

[Koey]

Homework Statement

A circle with a radius of (a/2) is bored through the centre of a sphere of radius a. Find the volume of the remaining solid.

Homework Equations

The Attempt at a Solution

I've been trying this for an hour now and I've been trying to find the remaining volume by finding the area of the cylinder bored + the little side bits on each side. I've just been getting really messy answers which aren't close to the actual answer.

Urgently need a hand :)

 

[justsof]

The easiest way to do this is in my opinion:
Set the sphere in the center of an xyz-axis system with the axis of the cilinder the y-axis. (for example)
Now consider just the first quadrant in the xy-axis system.
You should know that you can make a function of the curve you see ( f(x)= sqrt( (a^2)/4 - x^2 ) )

To find the volume of the cylinder inside the sphere you only have to rotate the function around the y-axis while integrating from 0 to a/4. Multiply by two and you have the total volume of the cylinder within the sphere.

I'm sure you will be able to solve it now ;).
Goodluck