Evaluate the iterated integral below.Note that it is necessary to switch the order of integration.

Question

Quizplus · Accepted Answer

Switching the order of integration, we have 
$$\int_0^{\pi/4}\int_y^{\pi/4}xy \cos x dx dy$$
Integrating with respect to $x$ first, we have 
\begin{align*}
\int_0^{\pi/4}\int_y^{\pi/4}xy \cos x dx dy&=\int_0^{\pi/4}\left(\sin x\cdot xy\Big|_y^{\pi/4}-\int_y^{\pi/4}\sin x y dxight)dy\
&=\int_0^{\pi/4}\left(\frac{\pi}{4}\cdot y\cdot \sin y-[-\cos x y]_y^{\pi/4}ight)dy\
&=\int_0^{\pi/4}\left(\frac{\pi}{4}\cdot y\cdot \sin y+\cos y\cdot y-\cos \frac{\pi}{4}\cdot yight)dy\
&=\left[\frac{\pi}{4}\cdot (-y\cos y +\sin y)-(\sqrt{2}-1)\cdot \sin \frac{\pi}{4}\cdot yight]_0^{\pi/4}\
&=\boxed{\frac{1}{2}\sqrt{2}+\frac{\pi}{16}-\frac{1}{\sqrt{2}}}
\end{align*}

Evaluate the Iterated Integral Below