Shortcuts to find a solution to a double integral

[Amaelle]

Homework Statement

Let T be the triangle of vertices (0,0),(2,2),(-4,0) . compute the integral ∫∫xydxdy over the region T

Relevant Equations

0<=y<=(x+4)/3
-4<=x<=2

I know the value of this integral is equal to 0, but I would like to see if there is any tricks to spot this answer using symmetries or even odd propreties?
Thanks in advance

 

[etotheipi]

This isn't a trick but I thought I'd just point out that the boundaries that you gave aren't right (i.e. yours is for a right angled triangle); you need to split the region of integration into $0 \leq y \leq \frac{x+4}{3}$, $-4 \leq x \leq 0$ as well as $x \leq y \leq \frac{x+4}{3}$, $0 \leq x \leq 2$.

 

[Amaelle]

yes thanks a lot , you are right the point is that I used a big a triangle with the following boundaries ( and calculated the integral over it) and then substructed the inetgral over a s smaller integral with the following boundaries
0<x<2 and 0<y<x