Question

The value of the integral $\displaystyle \iint_R xy\,dx\,dy$ over the region $R$, given in the figure, is ___________ (rounded off to the nearest integer).

 

Hint:

When the region is symmetric about the $y$-axis and the integrand is an odd function of $x$ (like $xy$), the double integral is zero without computation.

Answer 1

Step 1: Describe the region $R$.
The diamond has vertices at \((0,0)\) (intersection of \(y=x\) and \(y=-x\)), \((0,2)\) (intersection of \(y=x+2\) and \(y=-x+2\)), \((-1,1)\) (intersection of \(y=x+2\) and \(y=-x\)), and \((1,1)\) (intersection of \(y=-x+2\) and \(y=x\)). Hence \(R\) is symmetric about the $y$–axis. 
Step 2: Use symmetry of the integrand.
The integrand is \(xy\). For every point \((x,y)\in R\), the reflected point \((-x,y)\in R\) as well, and \[ xy + (-x)y = 0. \] Therefore, the contributions from symmetric pairs cancel. \[ \boxed{\,\iint_R xy\,dx\,dy=0\,} \]