\begin{gathered}\iint \limits _{S}\, sin(x+y)dx\, dy=\Big [\, y=x\; ,\; x=0\; ,\; y= \frac{\pi }{2} \; \to \; x=y=\frac{\pi}{2}\; \Big ]=\\\\=\int \limits _0^{\frac{\pi}{2}}\, dx\int \limits _{x}^{\frac{\pi}{2}}\, sin(x+y)\, dy= \int\limits^{\frac{\pi}{2}}_0\, dx \Big (-cos(x+y)\Big )\Big |\limits^{\frac{\pi}{2}}_{x}=\\\\= \int\limits^{\frac{\pi}{2}}_0\, dx \Big (-cos(x+\frac{\pi}{2})+cos2x\Big )= \int\limits^{\frac{\pi}{2}}_0 \Big (sinx+cos2x\Big )\, dx =\end{gathered}

=\Big (-cosx+\frac{1}{2}sin2x\Big )\Big |_0^{\frac{\pi}{2}}=-cos\frac{\pi}{2}+cos0-\frac{1}{2}sin\pi -\frac{1}{2}sin0=1=(−cosx+

2

1

sin2x)

0

2

π

=−cos

2

π

+cos0−

2

1

sinπ−

2

1 sin0=1