How to evaluate the integral $\iint_C \sin (x^2+y^2)\,\mathrm dx\,\mathrm dy$ I would like to evaluate
$$\iint\limits_C  \sin (x^2+y^2)\,\mathrm dx\,\mathrm dy$$
where $C$ is the first quadrant, i.e.
$$\int\limits_0^\infty\int\limits_0^\infty  \sin (x^2+y^2)\,\mathrm dx\,\mathrm dy$$
where
$$\int\limits_0^\infty \sin x^2\, dx=\int\limits_0^\infty \cos x^2\, dx=\frac{1}{2}\sqrt{\frac{\pi}{2}}$$

$$\begin{array}{c l}\int_0^\infty\int_0^\infty \sin(x^2+y^2)dxdy & = \int_0^\infty\int_0^\infty \sin(x^2)\cos(y^2)+\cos(x^2)\sin(y^2)dxdy \\[10pt] & =\int_0^\infty\sin(x^2)dx\int_0^\infty\cos(y^2)dy \\ & \qquad~~~+\int_0^\infty\cos(x^2)dx\int_0^\infty\sin(y^2)dy \\[10pt] & = 2\left(\frac{1}{2}\sqrt{\frac{\pi}{2}}\right)^2=\frac{\pi}{4}. \end{array}$$
