Triple Integral Calculator

What the calculator actually evaluates

This is a numerical calculator, not a symbolic integrator. It does not find antiderivatives or derive new bounds for spherical and cylindrical coordinates. If your region is a ball, cone, or other curved solid, first rewrite the problem into bounds and an integrand this rectangular midpoint method can sample, or use a symbolic triple-integral workflow.

Common uses

• Volume checks: use f(x,y,z) = 1 over a box to confirm the region volume.
• Mass estimates: enter a density function ρ(x,y,z) and integrate it over numeric bounds.
• Average value: divide the integral estimate by the volume of the box.
• Numerical verification: compare a hand-derived or textbook answer against a quick computational estimate.

Coordinate systems and volume elements

The interface accepts only x, y and z numeric ranges. Coordinate-system ideas are still useful for setting up the maths, but the final expression you enter must match the variables and limits the calculator samples.

Worked check: unit cube

For the unit cube and f(x,y,z) = x*y*z:

∭_{0≤x,y,z≤1} x*y*z dV
= (∫₀¹ x dx)(∫₀¹ y dy)(∫₀¹ z dz)
= (1/2)(1/2)(1/2)
= 1/8 = 0.125

This is a good test input because the bounds are rectangular and the expected answer is simple. If your numeric result is far from 0.125, increase the step count or check the expression syntax.

Accuracy tips

The midpoint rule usually improves as you increase the number of steps, but it is still an approximation. Smooth functions on a rectangular box behave best. Discontinuous functions, singularities, very narrow spikes, or regions that are not actually box-shaped can produce misleading values unless the problem is reformulated carefully.