# The integral $\int_{0}^{\sqrt\frac{\pi}{2}}\int_{x}^{\sqrt\frac{\pi}{2}}\sin(y^2) \, \, \mathrm{d}y \, \, \mathrm{d}x$

How do we go about computing the above integral? I figured we could write it as follows $$$$\int_{0}^{\sqrt\frac{\pi}{2}}\int_{0}^{\sin(x^2)}\sin(y^2) \, \,\mathrm{d}y \, \,\mathrm{d}x$$$$ But I really doubt that'd help in the first place. And am also not familiar with complex analysis :/. Would appreciate any assistance.

• Have you tried changing the order of integration from $dydx$ to $dxdy$, and adjusting the bounds accordingly? As a reminder, one way that this can be done is to begin by sketching the region of integration. Mar 3 '19 at 14:45

by chainging the order : $$\int_{y=0}^{y=\sqrt{\frac{\pi}{2}}} \int_{x=0}^{x=y} \sin(y^2)~dx~dy = \int y\sin(y^2)dy = \frac{-\cos(y^2)}{2}|_{0}^{\sqrt{\frac{\pi}{2}}} = \frac{1}{2}$$
• Could you explain how you adjusted your boundaries there? I don't see how $x$ runs from $0$ to $y$ Mar 3 '19 at 14:58