Calculate $\iint_S \sin(x+y)dxdy$

Question

Let S be the region bounded by the lines $y=x, x+y = \frac{\pi}{2}, y=0$ and calculate the double integral

$$\iint_S \sin(x+y)dxdy$$

I have sketched the region and get the following:

I'm unsure as to whether I integrate with respect to the lower triangle, or the top left triangle.

I have for the integral bounds the following when integrating from bottom: $$\int_{0}^{\frac{\pi}{2}}\int_{y}^{\frac{\pi}{2}-y}\sin(x+y)dxdy$$

Was this the correct region to integrate, otherwise - how do I tell?

Answer 1

Alternatively, per the symmetry of the four triangle regions $$\iint_S \sin(x+y)\>dxdy =\frac14 \int_0^{\frac\pi2}\int_0^{\frac\pi2} \sin(x+y)\>dxdy=\frac12$$