# Forced to use symmetry to solve this double integral?

1. Dec 4, 2013

### ainster31

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

http://i.imgur.com/d4ViHux.png

2. Relevant equations

3. The attempt at a solution

The author writes: "Now, using symmetry, we have..."

But what symmetry does the author use? Also, I got the integral as shown in the remark but why is it wrong?

2. Dec 4, 2013

### cepheid

Staff Emeritus
The symmetry that's used is only to integrate over only one semi-circle in the xy-plane, and not the other. That's why θ only ranges from 0 to π/2, not all the way to π. That's also why there is a factor of 2 in front of the integral: because the integral over one semi-circle should be exactly equal to the integral over the other one. This is because the portion of the hemisphere is that is above one semi-circle is equal in volume to the portion that is above the other: one portion is just the reflection of the other one across the y-axis. That is the symmetry.

3. Dec 4, 2013

### ainster31

Alright, but why is the integral shown in the remark incorrect?

4. Dec 4, 2013

### vela

Staff Emeritus
What did you get when you evaluated it? The integral isn't incorrect, but you have to be careful when evaluating it to get the correct result.

5. Dec 4, 2013

### haruspex

You have to be careful whenever you 'execute' a square root. √(x2) is not the same as x.