 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-iii/2/a/solution

x (θ) = f (θ) cos θ, y (θ) = f (θ) sin θ .

Differentiate using the product rule:

x^{'} = f^{'} cos θ - f sin θ, y^{'} = f^{'} sin θ + f cos θ .

Squaring and adding cancels the mixed terms, giving

(x^{'})^{2} + (y^{'})^{2} = (f^{'})^{2} + f^{2}

. The arc length is the integral of the speed, so

L = \int_{0}^{2 π} f (θ)^{2} + f^{'} (θ)^{2} d θ .

This assumes the parametrization is piecewise continuously differentiable and counts length with multiplicity if the curve is retraced.

 New to topics? Read the docs here!