Compute $$I=\iint_D\cos{y^2}dxdy,$$ where $D$ is the region bounded by the $y-$axis and the lines $y=x$, $y=\sqrt{\pi/2}$.
I've drawn the region and put the integration borders as follows:
So, $$I=\int_0^\sqrt{\frac{\pi}{2}}\left(\int_x^{\sqrt{\frac{\pi}{2}}}\cos{y^2}dx\right)dy.$$
But this involves some non-elementary functions so It can't be correct.
Your integral doesn't make much sense as you wrote it. It must be
$$\int_0^{\sqrt\frac\pi2}\int_x^{\sqrt\frac\pi2}\cos y^2\,\color{red}{\mathrm dy\,\mathrm dx}\stackrel{\text{Fubini}}=\int_0^{\sqrt\frac\pi2}\int_0^y\cos y^2\, \mathrm dx\,\mathrm dy=\int_0^{\sqrt\frac\pi2}y\cos y^2\,\mathrm dy=$$
$$=\left.\frac12\sin y^2\right|_0^{\sqrt\frac\pi2}=\frac12\left(\sin\frac\pi2-\sin 0\right)=\frac12$$