cant find the arc length fo the curve r = square root of (1 + cos 2 theta)?

Question

theta is greater or equal to 0 and less than or equal to pi sqr root of (2)

Answer 1

I will assume that this the equation is the equation of a
polar curve. A glance at any good calculus book shows
that the length of such a curve is given by
∫√ f(θ)² + f'(θ)² dθ

Now we need an identity from trig:
1 + cos 2θ = 2 cos²θ.
So r = √2 cos θ
r' = -√2 sin θ

And we get

L = ∫(0 to π√2) √2 cos²θ + 2 sin²θ dθ

L = 2π

Hope that helps!

Answer 2

Let l be the arc length.

r=√(1+cos2θ) = cosθ √2
dr/dθ = -sinθ √2

dl^2
=(rdθ)^2+(dr)^2
= (r^2+(dr/dθ)^2) (dθ)^2
= (1+cos2θ+2sin^2 θ) (dθ)^2
= 2 (dθ)^2

dl = √2 dθ

arc length( l )
= ∫dl
= ∫√2dθ [θ: 0...pi√2]
= 2pi