integration example
let the function be defined as:
\[
f(x,y) = e^{-(x^2 + y^2)} \cdot \sin(x)
\]
over the domain \(D = {(x,y) \in \mathbb{R} ^ 2 \mid -2 \le x \le 2, -2 \le y \le 2}\)
find:
1. The volume under \(|f(x,y)|\) over the entire domain.
$$
\iint_{D}|f(x,y)|\,dx\,dy
$$
2. The volume under \(|f(x,y)|\) restricted to the region where \(\sqrt{x^2 + y^2} > 0.5\).
\[\displaylines{
\iint_{R}|f(x,y)|\,dx\,dy \\
R = {(x,y) \in D \mid \sqrt{x^2 + y^2} > 0.5} \\
}\]
local numluau = require("@numluau")
local grid_x,grid_y = 1000,1000
local x = numluau.linspace(-2, 2, grid_x)
local y = numluau.linspace(-2, 2, grid_y)
local xv,yv = numluau.meshgrid(x,y)
local Z = numluau.exp(-(xv^2 + yv^2)) * numluau.sin(xv)
local dx = numluau.diff(x)[1]
local dy = numluau.diff(y)[1]
local volume = numluau.sum(numluau.abs(Z:flatten())) * dx * dy
local f = Z[numluau.greater(xv^2 + yv^2,0.5 ^ 2)]
local volume_contained = numluau.sum(numluau.abs(f:flatten())) * dx * dy
print("volume of |f(x,y)|:\n" .. volume:item() .. "\n")
print("volume of |f(x,y)| inside √(x² + y^²) > 0.5:\n" .. volume_contained:item())