Answer to Question #95176 in Physics for Manisha Nayak

Question

Using green's theorem evaluate the integral ellipse c (x2ydx-xy2dy) where c is the circle x2 +y2 =4 oriented counter clockwise.

Answer 1

We need to evaluate $\oint_C \bold F d \bold s$, where $\bold F = (P(x,y), Q(x,y)) = (x^2 y, -x y^2)$, and $C$ is the contour of the circle of radius $2$, centered at the origin.

According to Green's theorem, $\oint_C \bold F d \bold s = \iint_D (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial{y}})d A = \iint (-y^2 - x^2) d A = -\iint (x^2+y^2) dA$ , where we need to take the surface integral inside the circle.

Let us use polar coordinates: $x = r \cos \theta, y = r \sin \theta, d A = r dr d\theta$ , $0\leq r \leq 2, 0 \leq \theta \leq 2 \pi$. Hence, the integrand is $-\int_0^{2\pi} d \theta \int_0^2 r \cdot r^2 dr = -2 \pi \frac{r^4}{4}|_0^2 = -8 \pi$.