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Triple Integral Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A triple integral calculator is a mathematical utility that computes the integral of a function of three variables over a three-dimensional region. It evaluates volume integrals, mass, center of mass, and moments of inertia for solids bounded by specific surfaces. The calculator supports integration in Cartesian coordinates, cylindrical coordinates, and spherical coordinates. Students and physicists use this tool to solve complex multivariable calculus problems and verify analytic integration steps.

Solve ∫∫∫ f(x,y,z) dx dy dz over rectangular regions with numerical approximations.

Quick Answer

Calculate the triple integral of a three-variable function over a 3D region. Enter the function and the integration bounds for Cartesian, cylindrical, or spherical coordinates to see the result.

Use x, y, z and standard math operators (+, -, *, /, ^, sin, cos, etc.) · e.g. x * y * z

Result

Numerical Approximation

0.125

≈ 0.12500000

Integral Setup∫∫∫ (x * y * z) dx dy dz
x bounds[0, 1]
y bounds[0, 1]
z bounds[0, 1]
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How it works

What is a triple integral?

A triple integral evaluates a function of three variables over a three-dimensional region. It is the 3D equivalent of a standard integral. Just as a standard integral finds the area under a curve, a triple integral can find the volume of a 3D region (if the function is f=1) or the total mass of an object if the function represents variable density.

How iterated integration works

Thanks to Fubini's Theorem, a triple integral over a rectangular box can be computed as three nested single integrals. This is called an iterated integral.

∫∫∫ f(x,y,z) dV = ∫ [ ∫ [ ∫ f(x,y,z) dx ] dy ] dz

You work from the inside out:
1. Integrate with respect to x (treating y and z as constants).
2. Integrate the result with respect to y (treating z as constant).
3. Integrate the final result with respect to z.

Worked Example

Let's evaluate ∫∫∫ xyz dV over the region where x, y, and z all range from 0 to 1.

  • Inner Integral (dx):
    ∫ (xyz) dx from 0 to 1 = [ (x²/2)yz ] from 0 to 1 = (1/2)yz
  • Middle Integral (dy):
    ∫ (1/2)yz dy from 0 to 1 = [ (1/4)y²z ] from 0 to 1 = (1/4)z
  • Outer Integral (dz):
    ∫ (1/4)z dz from 0 to 1 = [ (1/8)z² ] from 0 to 1 = 1/8

Final Result = 0.125

Applications

  • Volume: If f(x,y,z) = 1, the integral gives the volume of the region.
  • Mass: If f(x,y,z) represents the density of a material at a point, the integral gives the total mass.
  • Center of Mass: By calculating moments in the x, y, and z directions using triple integrals, you can find the center of mass of a solid.

Disclaimer

For educational purposes. This calculator uses numerical methods to approximate the integral. Verify results for critical applications.

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Frequently asked questions

A triple integral is the extension of a single integral to three dimensions. While a single integral calculates the area under a curve, and a double integral calculates the volume under a surface, a triple integral calculates the 'hypervolume' or integrals of functions over a three-dimensional region. It is written as ∫∫∫ f(x,y,z) dV.

Triple integrals are evaluated using iterated integration, thanks to Fubini's Theorem. This means you evaluate three single integrals one after another, from the innermost integral to the outermost. For a rectangular region, the order of integration (e.g., dx dy dz or dz dy dx) does not affect the final result.

Fubini's Theorem states that if a function is continuous over a rectangular region in 3D space, its triple integral can be computed as an iterated integral, and the order of integration can be swapped without changing the result.

Triple integrals are widely used in physics and engineering to calculate volumes of 3D regions, the mass of an object with variable density, the center of mass, and the moment of inertia.

The expression 'dx dy dz' (often written as 'dV') represents an infinitesimally small volume element in Cartesian coordinates. The order specifies which variable is being integrated first. For example, dx dy dz means integrate with respect to x first, then y, then z.

Currently, this calculator supports numerical integration over rectangular regions (where the bounds for x, y, and z are constants). Integrating over complex regions requires variable bounds, which makes numerical evaluation much more complex.

Many functions do not have an elementary antiderivative, making exact symbolic integration impossible. Numerical integration (like the Riemann sum or Simpson's rule) approximates the value of the integral by breaking the region into small sub-regions and summing their contributions.

The result is an approximation. The calculator uses numerical integration with many subdivisions to provide a highly accurate result, but small rounding or approximation errors may be present depending on the complexity of the function.