Volume of region common to two intersecting cylinders

[FeDeX_LaTeX]

Homework Statement

Find the volume of the region common to the intersecting cylinders $x^2 + y^2 = a^2$ and $x^2 + z^2 = a^2$.

The Attempt at a Solution

I am totally stuck here. What do they mean when they say 'intersecting cylinders'? I've drawn graphs of circles of radius a, centred at the origin, in the x-y plane and the x-z plane. I've put them together and ended up with two identical circles cutting each other at right angles, and I don't see any cylinders... can anyone help me visualise this?

They have ended up with

$$8 \int_{x=0}^{a} \int_{y=0}^{\sqrt{a^2 - x^2}} z dy dx$$

I can understand where the limits of integration come from, but not the factor of 8, nor what is actually going on here...

 

[LCKurtz]

Look here for some pictures:

http://www.math.tamu.edu/~tkiffe/calc3/newcylinder/2cylinder.html

The 8 is because you are only calculating the first octant volume.

 

[FeDeX_LaTeX]

Thanks -- a picture really helped. I found it impossible to visualise.