Question

Find the exact length of the curve. Use a graph to determine the parameter interval.. r = (cos(θ/2))^2

Answer 1

cos(2θ) = 2cos^2θ - 1,

cos^2(θ/2)
Solving
cos^2θ = [1 + cos(2θ)]/2.

replacing θ with θ/2 gives.
cos^2(θ/2) = (1 + cosθ)/2.

length of a polar curve of the form r = f(θ) on [a, b] is given by:
L = ∫ √[r^2 + (dr/dθ)^2] dθ (from θ=a to b).
hope this helps