 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-iv/3/a/ii/solution

x = r (θ) cos θ

and

y = r (θ) sin θ

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

Squaring and adding cancels the mixed terms and gives

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

. Hence the arc length is

s = \int_{θ_{1}}^{θ_{2}} r (θ)^{2} + (\frac{d r}{d θ})^{2} d θ .

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