Converting to polar coordinates gives
[tex]\displaystyle\iint_R\sin(x^2+y^2)\,\mathrm dA=\int_{\theta=0}^{\theta=\pi/2}\int_{r=2}^{r=5}\sin(r^2)r\,\mathrm dr\,\mathrm d\theta[/tex]
[tex]=\displaystyle\pi\int_{r=2}^{r=5}2r\sin(r^2)\,\mathrm dr[/tex]
[tex]=\displaystyle\pi\int_{r^2=4}^{r^2=25}\sin(r^2)\,\mathrm d(r^2)[/tex]
[tex]=-\cos(r^2)\bigg|_{r^2=4}^{r^2=25}[/tex]
[tex]=\pi(\cos4-\cos25)[/tex]
