Triple Integral Calculator
Evaluate a triple integral of a monomial in x, y, z over a rectangular box region.
About Triple Integral Calculator
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
Worked example
With the default values below, Triple Integral Calculator returns ∭f dV of 0.12500000. Change any input above and every figure here is recalculated in your browser.
Inputs 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 axisaccepts 4–60
- 20
Results
- ∭f dV
- 0.12500000
Step by step
- 1Values used
- 2Triple integral
- 3∭f dV= 0.12500000
Assumptions and limits
- The result is derived only from the 8 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
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.