Evaluate a triple integral of a monomial in x, y, z over a rectangular box region.
Computes the triple integral ∭ f(x,y,z) dV over a rectangular box [xa,xb]×[ya,yb]×[za,zb] using 3D composite Simpson's rule. Each axis uses Simpson's 1/3 weights independently. Use a smaller grid (n=20) since the total evaluations grow as n³.
Triple integral
∭ f(x,y,z) dV ≈ (hx/3)(hy/3)(hz/3) Σᵢ Σⱼ Σₖ wᵢ wⱼ wₖ f(xᵢ, yⱼ, zₖ)
Triple integrals evaluate f at n³ points, so n=20 gives 9261 evaluations. This balances speed and accuracy.
This calculator handles rectangular boxes. For spherical or cylindrical regions, transform the integrand and adjust bounds accordingly.