Calculate double integral with boundaries

Question

I'm working on calculating the following integral in Mathematica:

$$\iint_D (30 x^2 + 49 xy + 20 y^2)^3 \, \mathrm{d} x \, \mathrm{d} y,$$

where $D$ is a region enclosed by four lines: $6x+5y=3$, $6x+5y=-3$, $5x+4y=1$, $5x+4y=-1$.

I know how to calculate double integrals, but I'm not sure how to integrate over a region with boundary lines. Is it possible to use Boole here?

Answer 1

reg = ImplicitRegion[{6 x + 5 y <=  3, 6 x + 5 y >=  -3,5 x + 4 y <= 1, 5 x + 4 y >=  -1}, {x, y}]  
Integrate[(30 x^2 + 49 x y + 20 y^2)^3, Element[{x, y}, reg]]
(*0*)

Hope it helps!