Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics III / 2 / a

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- Solution a

Solution

In Cartesian coordinates, a parametrization of the curve is

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.

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