Integration over ellipse

Question

$A=\{(x,y)\in \Bbb R^2\mid \frac{x^2}{a^2}+\frac {y^2}{b^2}=1\}$. Find $\int_A (\cos x)y\,dx+(x+\sin x)\,dy$.

Can someone please please give a methodological answer? Thanks a lot!

Answer 1

First you parametrize your ellipse : $A = (a\cos\theta, b\sin\theta) \in \mathbb{R}^2 \rvert \theta \in [0,2\pi[$. So you have : $$\left \lbrace \begin{array}{l} x = a\cos\theta\\ y = b\sin\theta \end{array} \right. \Rightarrow \left \lbrace \begin{array}{l} dx = -a\sin\theta \, d\theta\\ dy = b\cos\theta \, d\theta \end{array} \right. $$ Then rewrite your integral as : $$\oint_A y\cos x \, dx + (x + \sin x)\,dy = \oint_A x\,dy + d\left(\sin(x)y \right)$$ The last term vanishes because one integrates an exact differential on a closed loop. To compute the first term one uses the paremetrization of $A$ : $$\oint_A x\,dy = \int\limits_0^{2\pi} ab \sin^2\theta \, d\theta = \pi ab$$