Triple Integral Calculator
Evaluate a triple integral of a monomial in x, y, z over a rectangular box region.
Inputs
Enter function using x, y, z.
∭f dV
0.12500000
Step by step
Values used
f(x, y, z) = x*y*z; x lower = 0; x upper = 1; y lower = 0; y upper = 1; z lower = 0; z upper = 1; Grid points per axis = 20
Triple integral
∭ f(x,y,z) dV ≈ (hx/3)(hy/3)(hz/3) Σᵢ Σⱼ Σₖ wᵢ wⱼ wₖ f(xᵢ, yⱼ, zₖ)
∭f dV
= 0.12500000
How it works
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.
Formula
Triple integral
∭ f(x,y,z) dV ≈ (hx/3)(hy/3)(hz/3) Σᵢ Σⱼ Σₖ wᵢ wⱼ wₖ f(xᵢ, yⱼ, zₖ)
- hx
- (xb-xa)/n
- hy
- (yb-ya)/n
- hz
- (zb-za)/n
Frequently Asked Questions
Why is the default grid size smaller?
Triple integrals evaluate f at n³ points, so n=20 gives 9261 evaluations. This balances speed and accuracy.
What shapes can I integrate over?
This calculator handles rectangular boxes. For spherical or cylindrical regions, transform the integrand and adjust bounds accordingly.