- #1
ck99
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Homework Statement
Hi folks, I am sure this is very simple but there are not enough steps given in this calculation for my simple brain to get from the beginning to the end!
σ = ∫ (dσ/dΩ) = ∫ r2sin2θ (no integral limits given)
σ = 2∏r2 ∫ (1 - u2) du (integral from -1 to 1)
σ = 8∏r2 / 3
Homework Equations
u = cos θ
The Attempt at a Solution
I used Ω = sin θ dθ d∅ and first integrated ∅ from 0 to 2∏ to get
σ = ∫ dσ = r2 ∫ sin2θ dΩ
σ = 2∏ r2 ∫ sin2θ sin θ dθ
Use sin2θ = 1 - cos2θ to get
σ = 2∏ r2 ∫(1 - cos2θ) sin θ dθ
Let u = cos θ so du/dθ = - sin θ and dθ = -arcsin θ to get
σ = 2∏ r2 ∫(1 - u2) sin θ -arcsin θ du
I think sin and arcsin cancel to give
σ = 2∏ r2 -∫(1 - u2) du
σ = 2∏ r2 2u
From the answer that was given I have the integral limits running from -1 to 1 so the final term becomes [2 - (-2)] = 4 which gives
σ = 8∏ r2
Where am I going wrong please?