Double

This integral fails to converge. If we attempt to evaluate the infinite integral using polar coordinates, we obtain $\begin{array}{rcl} \displaystyle \int\int_{{x,y \ge0 \atop x^2 + y^2 \le R^2}} \sin(x^2+y^2)\,dx\,dy & = & \displaystyle \int_0^{\frac12\pi} \int_0^R \sin r^2\, r\,dr\,d\theta \\ & = & \displaystyle \tfrac12\pi\int_0^R r \sin r^2\,dr \\ & = & \displaystyle \tfrac12\pi \Big[-\tfrac12 \cos r^2 \Big]_0^R \\ & = & \displaystyle \tfrac14\pi\big(1 - \cos R^2\big) \end{array}$ which has no limit as $R \to \infty$ .

That we can calculate the infinite integral $\int_0^\infty \sin(x^2+y^2)\,dx \; = \; \int_0^\infty \big(\sin x^2 \cos y^2 + \cos x^2 \sin y^2\big)\,dx \; = \; \sqrt{\frac{\pi}{8}}\big(\sin y^2 + \cos y^2\big)$ so that the iterated integral $\int_0^\infty \left( \int_0^\infty \sin(x^2 + y^2)\,dx\right)\,dy \; = \; \tfrac14\pi$ exists is not relevant.