# Volume Integral

1. Apr 15, 2005

### Imo

"Find the volume of the region enclosed between the survaces $$z=x^2 + y^2$$ and $$z=2x$$"

I figured that the simplest way of doing this was to switch to a cylindrical co-ordinate system. Can someone check that the limits of integration are then
$$-\frac{\pi}{2}\leq \theta \leq\frac{\pi}{2}$$
$$0\leq\ r \leq 2\cos(\theta)$$
$$r^2\leq\ z \leq 2 r \cos(\theta)$$
(and the jacobian being r)

Thanks greatfully

2. Apr 15, 2005

### HallsofIvy

Staff Emeritus
z= 2x is a plane and forms the top of the figure. You are correct that you should use cylindrical coordinates. But z= x2+ y2 is a paraboloid. it's interesection with z= 2x is z= 2x= x2[/wsup]+ y2 or x2- 2x+ 1 + y2= (x-1)2+ y2 = 1 which, projected down in to the xy-plane is the circle with center (1,0) and radius 1. In cylindrical coordinates, x2+ y2= 2x is r2= 2rcos &theta; or
r= 2 cos &theta. THAT is the fomula you need.

3. Apr 15, 2005

### Imo

Unless I'm missing something (which is entirely possible), is that not what I have?

4. Apr 15, 2005

### learningphysics

Yes, I believe your limits of integration are correct.