Subtopic: Physical interpretations – line, surface and volume integrals
Question Statement:
Using Green's theorem in the plane, evaluate the line integral $\oint_C (x^2 - y) dx + (x + y^2) dy$ around the counterclockwise circular path $C: x^2 + y^2 = 4$. (Take $\pi = 3.1416$)
Answer Key:
25.13
Step-by-Step Solution & Explanation
By Green's theorem:
$\oint_C M dx + N dy = \iint_R \left( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \right) dA$
Here $M = x^2 - y$ and $N = x + y^2$:
$\frac{\partial N}{\partial x} = 1, \quad \frac{\partial M}{\partial y} = -1$
$\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = 1 - (-1) = 2$
$\oint_C M dx + N dy = \iint_R 2\, dA = 2 \times \text{Area}(R)$
Region $R$ is circle of radius $R = 2$, area $\pi (2)^2 = 4\pi$:
$\text{Integral} = 2 \times 4\pi = 8\pi = 8 \times 3.14159 \approx 25.132$