A question asks me to calculate the following integral using NIntegrate:
Calculate the double integral ... where D is the area bounded by the four lines ...
With the following code I only manage to get zero as an answer, which is wrong:
f[x_, y_] := (30*x^2 + 37*x*y + 10*y^2)^2
NIntegrate[f[x, y] Boole[6x + 5y == 9 && 6x + 5y == -9 && 5x + 2y == 1 &&
5x + 2y == -1], {x, -Infinity, Infinity}, {y, -Infinity, Infinity}]
What should I do instead?
You want the region of integration to be defined by inequalities, rather than equalities. Your region can be described as follows:
region = ImplicitRegion[Abs[6*x + 5*y] < 9 && Abs[5 x + 2 y] < 1, {x, y}]];
RegionPlot[region, AspectRatio -> Automatic]
With that definition, you can then write:
ClearAll[f]
f[x_, y_] := (30*x^2 + 37*x*y + 10*y^2)^2
NIntegrate[
f[x, y],
{x, y} ∈ region
]
(* Out: 24.9231 *)
In this case, you can also obtain a symbolic answer, just by replacing Integrate for NIntegrate above (it returns 324/13).