Evaluating double integral by polar coordinates

Question

• $\displaystyle S$ is an area in the first quadrant in a Cartesius plane which is bounded by circle $\displaystyle x^{2} + y^{2} = 1$ and lines $\displaystyle y = 0$ and $\displaystyle x = 0$.
• Calculate $\displaystyle\iint\sin\left(\pi\left[x^{2} + y^{2}\right]\right)\,\mathrm{d}A$

I don't really understand the topic but I thought that we should convert it to polar coordinates and that's it; but then the area is also bounded by the lines $\displaystyle x = y = 0$ so there must be something else.

Help me please, thank you.

Answer 1

I also don't understande which region is that. But if it turns out to be the region where $x,y\geqslant0$, then you should compute$$\int_0^{\frac\pi2}\int_0^1\sin(\pi\rho^2)\rho\,\mathrm d\rho\,\mathrm d\theta$$because $\theta\in\left[0,\frac\pi2\right]\iff x,y\geqslant0$.