# Surface integrals - parametrizing a part of a sphere

1. Apr 9, 2014

### Feodalherren

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

Find the area of the part of the sphere x^2 + y^2 + z^2 = 4z
that lies inside the paraboloid x^2 + y^2 = z

2. Relevant equations

3. The attempt at a solution
I solved for the intercepts and found that they are z=0 and z=3.
The sphere is centered two units in the z-direction above the origin.

Hence I wanted to switch to spherical coordinates and got:

0≤Θ≤2∏
ρ=2

Now, since we know that the sphere is centered on (0,0,2) we can take find the angle from the Z axis easily.

(∏/3)≤Φ≤∏.

The Surface Area of this portion of the sphere should then become
$\int^{2∏}_{0} \int^{∏}_{∏/3} 4SinΦ dΦdΘ$
$\int^{2∏}_{0} \int_{∏/3}_{∏} 4SinΦ dΦdΘ$

I get 12∏, which is incorrect. The correct answer is 4∏. Where am I going wrong?

Last edited: Apr 9, 2014
2. Apr 9, 2014

### Feodalherren

That itex thing never works for me either :/

3. Apr 9, 2014

### Feodalherren

Nevermind I solved it. I was taking the wrong portion of the curve.

4. Apr 9, 2014

### LCKurtz

In the tex, don't use special characters; instead use the tex code for them. Quote this to see how I changed your code.$$\int_0^{2\pi}\int_{\pi/3}^{\pi} 4\sin\theta d\phi d\theta$$