As stated in the title, I'm trying to calculate $$ \iint\limits_H \, (x+y) \mathrm{d} A $$
Where $H$ is the area of the cardioid $r=1+\cos (\theta) $.
I have calculated the area of $H$ to be $$ \frac{1}{2}\int_{\theta=0}^{2\pi} (1+\cos(\theta))^2 \mathrm{d}\theta = 2 \cdot \frac{1}{2}\int_{\theta=0}^{\pi} (1+\cos(\theta))^2 \mathrm{d}\theta = 1\frac{1}{2}\pi.$$
However, when trying to calculate $ \iint\limits_H \, (x+y) \mathrm{d} A $ and using the fact that $x=r \cos\theta$ and $y=r \sin\theta$ I try to rewrite $r \cos\theta + r \sin\theta$ to something that does not include the variable $r$, but I can't seem to find what to rewrite it to or what limits to use for the integral. Any help in solving this is greatly appreciated.
Best Answer
$$ \iint\limits_H \, (x+y) \mathrm{d} A $$
Convert the integral to polar coordinates. Let $x=r\cos\theta$ and $y=r\sin\theta$. The determinant of the Jacobian matrix is given by $r$.
Thus, $dA=dxdy=rdrd\theta$
$$x+y=r(\cos\theta+\sin\theta)$$
The integral can now be written as, $$\int_{\theta=0}^{2\pi}\int_{r=0}^{1+\cos\theta}r^2(\sin\theta+\cos\theta)drd\theta$$