What functions can I enter?

Any expression using x, y, and z with standard operators: polynomials (x^2 + y^2 + z^2), trig (sin(x)*cos(y)*z), exponentials (exp(x+y+z)), logarithms (ln(x*y*z)), absolute value (abs(x-z)), and more. Use * for multiplication explicitly (2*x, not 2x). Raise to a power with ^ (x^3).

How accurate is the result?

The midpoint Riemann sum with n=30 (27,000 sub-boxes) gives accuracy within roughly 0.1–1% for smooth functions over moderate domains. Increasing n to 50 (125,000 points) improves accuracy significantly but takes a moment longer. For highly oscillatory or near-singular functions, use a larger n.

What does the grid size n control?

n is the number of equal sub-intervals along each axis, so the total number of sub-boxes evaluated is n³. A grid of n=20 evaluates 8,000 points; n=50 evaluates 125,000 points. Larger n reduces step error but increases computation time.

Why does my result show NaN or Infinity?

NaN means the function returned an undefined value at one or more grid points — for example, log of a negative number, division by zero, or a syntax error. Check that your function syntax is valid and that the domain avoids singularities.

Can I integrate over non-box regions?

This tool supports rectangular box regions only. To integrate over a non-box region, set f(x,y,z) = 0 outside the desired region using a conditional expression, or decompose the region into boxes and sum the results separately.

What is the difference between this and the double integral tool?

The double integral tool evaluates ∬f(x,y) dA over a 2D rectangle. This tool adds a third dimension and evaluates ∭f(x,y,z) dV over a 3D box. Use this when your function depends on three variables.

Triple Integral Numeric Solver

Name: Triple Integral Numeric Solver
Availability: InStock
Author: Nham Vu

Enter f(x,y,z), set bounds for all three axes, and get a numeric approximation of the triple integral using a 3D midpoint Riemann sum.

Configure Integral

f(x, y, z)

Use x, y, z as variables. Use * for multiply, ^ for power.

x bounds [a, b]

a (x lower)

b (x upper)

y bounds [c, d]

c (y lower)

d (y upper)

z bounds [e, g]

e (z lower)

g (z upper)

Grid size n (sub-intervals per axis)

Examples

Set the function and bounds, then click Calculate.

About the method

The tool divides the box [a,b]×[c,d]×[e,g] into n³ sub-boxes of equal size. For each sub-box, it evaluates f at the midpoint (x_m, y_m, z_m) and multiplies by the volume Δx·Δy·Δz. The total is the sum of all products — the 3D midpoint Riemann sum. For smooth functions, error decreases as O(h²) where h = max(Δx, Δy, Δz), so doubling n roughly quarters the error. Note that n=50 already evaluates 125,000 points; keep n modest for fast results on simple functions.

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Summary

Enter f(x,y,z), set bounds for all three axes, and get a numeric approximation of the triple integral using a 3D midpoint Riemann sum.

How it works

Enter the function f(x,y,z) using standard notation (e.g. x*y*z, sin(x+y+z), x^2 + y^2 + z^2).
Set the x lower and upper bounds (a, b).
Set the y lower and upper bounds (c, d).
Set the z lower and upper bounds (e, g).
Choose the grid size n (sub-intervals per axis; n³ total points evaluated).
Click "Calculate" to run the 3D midpoint Riemann sum and see the result.

Use cases

Verify triple integral results from calculus textbooks or homework.
Estimate volume of a solid or mass of a 3D object with known density function.
Compute 3D probability integrals for joint density functions of three variables.
Explore how box bounds affect the integral value interactively.
Check symbolic CAS output against a numeric approximation.
Demonstrate iterated integration over a box domain in a classroom setting.
Approximate center-of-mass integrals in physics problems.

Frequently Asked Questions

Last updated: 2026-07-26 · Reviewed by Nham Vu