What will be the value of $\int_\gamma x\cdot n(x) \, ds(x).$

Question

Let $x=(x,y)\in \mathbb R^2$, $n(x)$ denote the unit outward normal to the ellipse $\gamma$ whose equation is given by $\frac{x^2} 4 +\frac{y^2} 9 = 1$ at the point $x$ on it.

What will be the value of $\displaystyle\int_{\gamma}x\cdot n(x)\,ds(x)\text{ ?}$

Answer 1

Hint. Use the planar version of the Divergence Theorem: $$\int_{\partial D} (v_1,v_2) \cdot n \, ds=\int_D \left(\frac{\partial v_1}{\partial x} +\frac{\partial v_2}{\partial y} \right)\, dxdy.$$ In particular take a look to this example.