Chapter 13.1, Problem 38E

Symmetry Evaluate the following integrals using symmetry arguments. Let $R=\left\{\left(x,y\right):=-a\le x\le a,-b\le y\le b\right\}$, where a and b are positive real numbers.

a.     ${\displaystyle \underset{R}{\iint }xy{e}^{-\left(x^2+y^2\right)}}$

b.     ${\displaystyle \underset{R}{\iint }\frac{\sin \left(x-y\right)}{x^2+y^2+1}}dA$