Triple Integral Calculator

A triple integral ∫∫∫_V f(x,y,z) dV computes a quantity (mass, charge, average value, hyper-volume) over a 3D region V. For a rectangular box [a,b] × [c,d] × [e,g] it can be evaluated as a triply nested iterated integral:

∫_a^b ∫_c^d ∫_e^g f(x,y,z) dz dy dx

Innermost integral first (with x and y treated as constants), then middle, then outer.

This calculator does the integral two ways and shows both. Because all three integrand forms below are separable powers, the closed form is available, and that exact value is what the headline reports. Alongside it sits the result of Simpson’s rule in three dimensions: for each grid x value it integrates over y, and for each (x, y) it integrates over z. With n = 20 subintervals per dimension that is (n+1)³ = 9,261 function evaluations, fast for any modern browser.

Comparing the two is the interesting part. Simpson’s rule is exact for polynomials up to degree 3, so for small exponents the numerical and exact answers agree to the last digit. Push an exponent to 8 and a visible gap opens: the numerical answer for ∫∫∫ x⁸yz over the unit cube comes out 0.027781 against an exact 1/36 = 0.027778. That difference is the price of numerical integration, and seeing it on a problem you can check by hand is worth more than reading about it.

Three integrand forms are available:

Sum: f(x,y,z) = ax^p + by^q + cz^r. Each term integrates separately over its own variable times the lengths of the other two intervals. Product: f(x,y,z) = a · x^p · y^q · z^r. Splits cleanly into three 1D integrals: a · ∫x^p dx · ∫y^q dy · ∫z^r dz. Constant: f = a. The integral is just a × (volume of the box).

Worked example: ∫_0^1 ∫_0^1 ∫_0^1 xyz dz dy dx. Splits as (∫_0^1 x dx)(∫_0^1 y dy)(∫_0^1 z dz) = (1/2)(1/2)(1/2) = 1/8.

Common physical interpretation: if f(x,y,z) is a density (mass per unit volume), the triple integral gives total mass. If f is unity, it gives volume. If f is a function and you divide by volume, you get the average value of f over the region.

For non-rectangular regions (spheres, tetrahedra, cylinders) you would either set up variable inner bounds, or change variables (spherical, cylindrical, or general substitution with the Jacobian). This calculator handles the rectangular box case, which is by far the most common in textbook problems and the foundation for all the others.