what is the value of $\int_{\lambda}(n_1(x,y)x+n_2(x,y)y)ds.$

Question

If $n=(n_1(x,y)+n_2(x,y))$ is the outward unit normal at the point $P=(x,y)$ lying on the curve $\lambda$ which is $x^2+4y^2=4$, Then what is the value of

$\int_{\lambda}(n_1(x,y)x+n_2(x,y)y)ds.$

Answer 1

Use Green's Theorem or 2D divergence theorem. This integral is

$$\oint \vec{F}\cdot d\vec{s}=\iint\bigtriangledown\cdot F dS $$ where $\vec{F}=(x,y)$. So you only need to find the divergence of $\vec{F}$, which is $2$, and the area of the ellipse $\pi ab$.