How could you integrate the function $f(x,y) = x^2 + y^2$ over the triangle with vertices $(0,0)$, $(0,1)$ and $(1,0)$?
I define the set $D = \{(x,y)\; |\; 0\leq x\leq 1 \text{ and } 0\leq y\leq x\}$ and then calculate
$\int_0^1 \int_0^x x^2 + y^2 \; \mathrm{d}y \; \mathrm{d}x = \frac{1}{3},$
but apparantly the answer is $\frac{1}{6}$.