I came up with this...
Is this correct?
Hmmm I'm finding instead of
Also, there is that coefficient, 2, disturbing me. We once calculated the dxdydz, and my friend found this coefficient. I wasn't able to find the mistake in either of our two computations.
But in the wikipedia, there isn't this coefficient.
As for the boundaries of your integral, it depends on how you define and
In regards to the question, I'm confused about where the 2 is coming from as well...I will ponder this a little more. Any insight into how you came up with the 2? I'm just not seeing it.
So, then the integral becomes
Perhaps your two came from a trig identity, I'm too tired to think about it..if it came from an identity then it's fine.
Also, I have my integration in a little bit different order than you, but since we are in spherical coordinates and the sphere is centered at the origin we can flip the integration limits arbitrarily.
I think this is all correct...good luck