Triple Integral Calculator

A triple integral extends the concept of a single and double integral to three dimensions. It integrates a function f(x, y, z) over a three-dimensional region, and is written as ∭f(x,y,z) dV. Triple integrals are used to compute volumes, masses, centroids, and other quantities over 3D regions.

Triple integrals have many practical applications. They can calculate the volume of a 3D solid, the total mass of an object with variable density, the center of mass or moment of inertia of a 3D body, and the total charge within a region of space in electrostatics or fluid dynamics problems.

To set up a triple integral, identify the region of integration in three-dimensional space and determine the bounds for each variable. Write the integrand f(x, y, z), then integrate step by step with respect to the innermost variable first, holding the others constant, and work outward through the remaining variables.

The order of integration specifies which variable is integrated first, second, and last (e.g., dx dy dz means integrate with respect to x first, then y, then z). For a rectangular region with constant bounds, the result is the same regardless of the order chosen (Fubini's theorem). For variable bounds, the correct order depends on the geometry of the region.

Triple integrals can be evaluated in Cartesian coordinates (x, y, z), cylindrical coordinates (r, θ, z), or spherical coordinates (ρ, θ, φ). Cylindrical and spherical coordinates often simplify integration over regions with circular or spherical symmetry. This calculator uses Cartesian coordinates with numerical evaluation.

This calculator uses numerical approximation (Gaussian quadrature / nested Simpson's rule) to evaluate the definite triple integral over the specified rectangular region. It divides each variable's range into small intervals and sums the weighted function values. The result is highly accurate for well-behaved continuous functions.

You can enter standard algebraic expressions using x, y, and z with operators +, -, *, /, and ^ for exponentiation. Trigonometric functions (sin, cos, tan), exponential (exp), square root (sqrt), and logarithmic (log) functions are also supported. Make sure to write expressions like x*y rather than xy for multiplication.

One of the most common mistakes is setting up incorrect bounds for the region of integration, especially when the region is not rectangular. For non-constant bounds (where one variable's limits depend on another), symbolic solvers are better suited. Also, forgetting the Jacobian when converting to cylindrical (r) or spherical (ρ² sin φ) coordinates is a frequent error.