 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/ia/paper-3/3c/i/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

Differentiate the parametrized curve to obtain

x^{'} (t) = (1 - t^{2}, 2 t, 1 + t^{2})

. Its speed obeys

∣ x^{'} (t) ∣^{2} = (1 - t^{2})^{2} + 4 t^{2} + (1 + t^{2})^{2} = 2 (1 + t^{2})^{2} .

Since

1 + t^{2} > 0

L = \int_{0}^{1} 2 (1 + t^{2}) d t = \frac{4 2}{3} .

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