# Setting up limits of integration for multiple integral

1. Apr 19, 2013

### dustbin

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

I need to find the volume of the region bounded by

$$(x-1)^2 + y^2 =1 \ \ \text{and} \ \ x^2+y^2+z^2=4 \ .$$
But I only need help setting up the limits of integration.

2. Relevant equations

The typical cylindrical change of variables.

3. The attempt at a solution

I have $0 \leq r \leq 2\cos\theta, \ -\sqrt{4-r^2} \leq z \leq \sqrt{4-r^2}, \ \text{and} \ -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}.$ Then the volume is given by
$$\int\limits_{-\pi/2}^{\pi/2}\int\limits_0^{(2\cos\theta)}\int\limits_{(-\sqrt{4-r^2})}^{(\sqrt{4-r^2})} dz\,(r\,dr)\,d\theta \ .$$

2. Apr 19, 2013

### haruspex

Pls define your cylindrical coordinate frame.

3. Apr 19, 2013

### HallsofIvy

Staff Emeritus
Your integral is over a cylinder with center at (0, 0). The cylinder of the problem has center at (1, 0).

4. Apr 19, 2013

### dustbin

@haruspex: Sorry, but I do not know what you mean by cylindrical coordinate frame.

@HallsofIvy: I thought that taking $0 \leq r \leq 2\cos\theta$ with $\theta\in(-\pi/2, \pi/2)$ made it so that I would be integrating over the projection of the cylinder onto the $x,y$ plane as a circle of radius 1 centered at (1,0). The region is bound by $\pm\sqrt{4-r^2}$ on $z$. Can I get some insight into how I can fix my limits of integration?

5. Apr 19, 2013

### SammyS

Staff Emeritus
That integral looks fine to me.