Calculus 2, Lec 15E: Arc Length of a Polar Curve (Elliptic Function Needed to Evaluate for Limaçon)

In Calculus, we represent a polar curve r=f(θ) with parametric equations as x=f(θ)cos(θ), y=f(θ)sin(θ). The integrand of the arc length integral is then ((dx/dθ)^2+(dy/dθ)^2)^(1/2). This ends up simplifying to ((f(θ))^2 + (f'(θ))^2)^(1/2) because of trigonometric identities. When we apply this to the example f(θ)=2+4cos(θ), which is a limaçon, we get an integral that cannot be exactly evaluated without the use of a special function, called an elliptic function.

Links and resources
===============================

(0:00) Arc length integral formula in terms of parameterization x(θ) and y(θ) of polar curve r=f(θ)
(2:02) Simplify the formula to relate it to r=f(θ) and dr/dθ=f'(θ)
(6:06) Arc length integral for limaçon r=f(θ)=2+4cos(θ)
(7:42) Mathematica shows an Elliptic function is necessary
(9:46) Approximate arc length of limaçon r=f(θ)=2+4cos(θ)

AMAZON ASSOCIATE
As an Amazon Associate I earn from qualifying purchases.