# Another Double Integral

1. Jul 4, 2013

### FeDeX_LaTeX

1. The problem statement, all variables and given/known data

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

3. 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...

2. Jul 4, 2013

3. Jul 4, 2013

### FeDeX_LaTeX

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