Arc Length, Polar Curves?

Question

r = 2(1-cosӨ)

Find the arc length of the polar curve ^ from 0 to п (pi)

formula for Arc Length = INT 0 --> п (pi) [f'(Ө)^2 + f(Ө)^2]^1/2 *dӨ

PLEASE show work! I keep getting back to the original problem. HELP!

Answer 1

Let me try.
r = 2(1-cosӨ)
dt/dӨ = 2sinӨ
∫[0 --> п] √(r^2 + (dr/dӨ)^2) dӨ
= ∫[0 --> п] √(4 - 8cosӨ + 4(cosӨ)^2 + 4(sinӨ)^2) dӨ
= 2 ∫[0 --> п] √(2 - 2cosӨ) dӨ
= 4 ∫[0 --> п] sin(Ө/2) dӨ
= -8 cos (Ө/2)|[0 --> п]
= 8

Answer 2

For parabolic coords (A,B) and cartesian (x,y) x = AB y = (a million/2 of)*(B^2-A^2) arc length: s = r(theta) r=sqrt(x^2+y^2) theta = arctan(y/x) so, arc length = sqrt(x^2+y^2)*arctan(y/x) = sqrt((AB)^2+(a million/2)*(B^2-A^2)y^2)* arctan((a million/2)*(B^2-A^2)y/AB) which will additionally be simplified to tidy it up (sorry havent gained time at mo!)