Surface between $f(x,y)=y+\sqrt{x^2+y^2}$ and $R={(x,y) \in x^2+y^2\leq 4} \wedge y\geq |x|$

Question

Find the $$\iint_R f(x,y)\,dA$$

I don't know how to work with the integrals. Any help is welcome.

Answer 1

I suspect the surface-integral tag is mistakenly included here, and that the integral is just a standard double integral over $R$.

Rewrite the region $R$ in polar coordinates:

$$R'=\left\{(r,\theta) \mid \frac\pi4\le\theta\le\frac{3\pi}4 \land 0\le r\le2\right\}$$

Then

$$\iint_R f(x,y)\,\mathrm dA = \iint_{R'} rf(r\cos\theta,r\sin\theta)\,\mathrm dr\,\mathrm d\theta = \int_{\frac\pi4}^{\frac{3\pi}4}\int_0^2 r^2(\sin\theta+1)\,\mathrm dr\,\mathrm d\theta$$

Can you take it from here?