 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/ia/paper-3/9a/ii/solution

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

On the first leg

y = 0

and

x

decreases from one to zero; on the second

x = 0

and

y

increases from zero to one. Adding their line integrals gives

I_{2} = α \int_{1}^{0} x^{2} d x + β \int_{0}^{1} y^{2} d y = \frac{β - α}{3} .

The negative sign in the first term follows from the direction of traversal.

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