Triple Integral Calculator: Solve Iterated Integrals Free

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This free triple integral calculator evaluates an iterated integral ∬ f(x, y, z) dV over a region you define. Enter the integrand and the limits — the inner limits may depend on the outer variables — and it returns the value, so it works for boxes, tetrahedra and other solids, not just rectangular regions.

x fromto
y fromto
y limits may use x — e.g. 0 to 1-x
z fromto
z limits may use x and y — e.g. 0 to 1-x-y
Triple integral
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Enter the integrand and limits, then press Calculate.
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What the triple integral calculator computes

A triple integral $\iiint_E f(x,y,z)\,dV$ adds up the values of a function over a three-dimensional region $E$. When $f=1$ it gives the volume of the region; with other integrands it gives totals like mass, charge, or average value. This calculator evaluates the integral as an iterated integral — integrating in $z$, then $y$, then $x$ — using fast numerical integration, so you get an accurate value even when no neat closed form exists.

How to set up the limits

The order matters. Working from the inside out, each variable's limits may depend only on the variables outside it:

VariableLimits may useExample
z (inner)x and y0 to 1−x−y
y (middle)x only0 to 1−x
x (outer)constants only0 to 1

For a simple box, all six limits are constants. For a tetrahedron or other sloped region, the inner limits are expressions — exactly what the inputs above allow.

Worked example: the volume of a tetrahedron

The solid bounded by the planes $x=0,\,y=0,\,z=0$ and $x+y+z=1$ has volume $$\int_{0}^{1}\!\int_{0}^{1-x}\!\int_{0}^{1-x-y} 1 \; dz\,dy\,dx = \tfrac{1}{6} \approx 0.1667.$$ Tap "volume of a tetrahedron" above to see the calculator return exactly that.

Fubini's theorem and order of integration

Fubini's theorem says that for a well-behaved function you may integrate in any order and get the same answer — $dz\,dy\,dx$, $dx\,dy\,dz$, and the rest all agree. In practice you pick the order that makes the limits simplest. This tool uses the common $dz\,dy\,dx$ order; to use another order, relabel your variables accordingly.

⚠ Cylindrical and spherical coordinates. For cylinders and spheres, converting to cylindrical $(r,\theta,z)$ or spherical $(\rho,\theta,\varphi)$ coordinates makes the limits far simpler — but remember the volume element changes: $dV = r\,dr\,d\theta\,dz$ or $dV = \rho^2\sin\varphi\,d\rho\,d\varphi\,d\theta$. Include that factor in your integrand here.

What triple integrals are used for

  • Volume of a 3-D region (integrate $f=1$).
  • Mass and density — integrate a density function over a solid.
  • Center of mass and moments of inertia in physics and engineering.
  • Probability — integrating a joint density over a 3-D region gives a probability, a idea that underpins multivariate statistics and machine learning.

For the single-variable building block, see our integral calculator. For the formal definition, the multiple integral article on Wikipedia is a solid reference.

Frequently asked questions

What is a triple integral? It is an integral of a function of three variables over a 3-D region, written $\iiint_E f(x,y,z)\,dV$. With $f=1$ it gives the region's volume.

How do I find the limits of a triple integral? Work from the inside out: the inner variable's limits can depend on the outer variables, the middle variable's limits can depend on the outermost, and the outer limits are constants.

Can the limits be variables? Yes. Enter expressions such as 1-x for the y-limits and 1-x-y for the z-limits to integrate over non-box regions.

How accurate is the calculator? It uses nested Simpson's-rule integration, which is accurate to several decimal places for smooth integrands.

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