Arc length of a polar curve

Find the exact length of the polar curve r=cos^2(theta/2)

Pleeeease help!

Note that the entire curve is plotted for θ in [0, 2π].

Plot:
http://www.wolframalpha.com/input/?i=pol…
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So, the arc length ∫ √(r^2 + (dr/dθ)^2) dθ equals
∫(θ = 0 to 2π) √[(cos^2(θ/2))^2 + (-cos(θ/2) sin(θ/2))^2] dθ
= ∫(θ = 0 to 2π) |cos(θ/2)| √[cos^2(θ/2) + sin^2(θ/2)] dθ
= ∫(θ = 0 to 2π) |cos(θ/2)| * 1 dθ
= ∫(w = 0 to π) |cos w| * 2 dw, letting w = θ/2.
= 2 * ∫(w = 0 to π/2) +(cos w) * 2 dw, via symmetry
= 4 sin w {for w = 0 to π/2}
= 4.

I hope this helps!