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 uses Simpson’s rule in three dimensions. For each grid x value, it integrates over y at each x; for each (x,y) it integrates over z. With n=20 subintervals per dimension the calculation requires (n+1)³ = 9261 function evaluations, which is fast for any modern browser.

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.