 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ia/paper-2/3b/b/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

Green theorem states that for a positively oriented planar region

D

\oint_{\partial D} P d x + Q d y = \iint_{D} (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) d A .

Here

P = x y^{2} - λ y

and

Q = x^{2} y

, so

Q_{x} - P_{y} = 2 x y - (2 x y - λ) = λ .

The stated orientations make

C_{1} \cup C_{2}

the positive boundary of the region inside the radius-

a

circle and outside the radius-

b

circle. Its area is

π (a^{2} - b^{2})

regardless of the inner circle's center. Therefore

\int_{C_{1} \cup C_{2}} P d x + Q d y = πλ (a^{2} - b^{2}) .

 New to topics? Read the docs here!