(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find σ , the differential cross section, starting from the expression below and integrating over solid angle Ω

2. Relevant equations

dσ/dΩ = r^{2}sin^{2}θ

3. The attempt at a solution

dσ = r^{2}sin^{2}θ dΩ

I remember that dΩ = sinθ dθ dμ

and doing the μ integral from 0 to 2∏ gives dΩ = 2∏ sinθ dθ

substitute in to get dσ = 2∏ r^{2}sin^{2}θ sinθ dθ

Now, the next step in the textbook is to substitute u = cos θ and write

σ = 2∏r^{2}∫(1-u^{2})du with u = cos θ and integration limits from -1 to 1.

I have played around with this step for a while and I know the trig identity sin^{2}θ = 1 - cos^{2}θ, but I still don't get how to do this step properly. I think I'm suppose to put a du/dθ term in there, which is a -sinθ term, but I can't see how they get to the final expression in the book, which is

σ = 8∏/3

If anyone can help me work through these steps in detail I would really appreciate it; I should know how to do this by now (4th year physics MSc student!) but it's like my brain switches off every time I see the ∫ symbol and I have to learn everything from scratch every term . . . :(

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# Homework Help: Integral of differential cross section over solid angle

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