Integrate[
Boole[{ -2 <= 2 x + 7 y <= 2 && -2 <= 3 x + 6 y <= 2}](6 x^2 +
33 x*y + 42 y^2)^4, {x, -∞, ∞}, {y, -∞, ∞}]
Integrate[(6 x^2 + 33 x*y + 42 y^2)^4, {x, y} ∈
ImplicitRegion[{ -2 <= 2 x + 7 y <= 2 && -2 <= 3 x + 6 y <= 2}, {x,
y}]]