Mathematics · Calculus · Integral Calculus
Triple Integral Calculator
Computes the triple integral of a function f(x, y, z) over a rectangular box domain by evaluating iterated integrals with specified limits.
Calculator
Formula
f(x, y, z) is the integrand function evaluated at point (x, y, z). x₁ and x₂ are the lower and upper limits of integration along the x-axis. y₁ and y₂ are the lower and upper limits along the y-axis. z₁ and z₂ are the lower and upper limits along the z-axis. dV = dx dy dz is the volume element. The integral is computed from the innermost (x) outward to the outermost (z) dimension. For a constant integrand f = 1, the result equals the volume of the rectangular box.
Source: Stewart, J. Calculus: Early Transcendentals, 8th Edition. Cengage Learning, Section 15.7.
How it works
A triple integral extends the concept of single and double integration into three dimensions. Instead of finding the area under a curve or the volume under a surface, a triple integral accumulates the value of a function f(x, y, z) throughout a three-dimensional region V. When f(x, y, z) = 1, the result is simply the volume of the region. When f represents density (mass per unit volume), the result is the total mass of the solid. In electromagnetism, it can represent total charge; in fluid dynamics, total flux through a volume.
For a rectangular box domain — where x ranges from x₁ to x₂, y from y₁ to y₂, and z from z₁ to z₂ — Fubini's Theorem guarantees the integral can be evaluated as an iterated integral in any order. The formula ∭ f(x,y,z) dV = ∫[z₁ to z₂] ∫[y₁ to y₂] ∫[x₁ to x₂] f(x,y,z) dx dy dz is computed from inside out: first integrate over x (treating y and z as constants), then integrate the result over y, then over z. For separable functions of the form f(x,y,z) = g(x)·h(y)·k(z), the triple integral factors into a product of three independent single integrals, greatly simplifying computation.
Practical applications of triple integrals span nearly every quantitative field. Mechanical engineers use them to find moments of inertia of solid objects for rotational dynamics. Aerospace engineers compute gravitational potential fields. Physicists evaluate probability densities in quantum mechanics using three-dimensional wavefunctions. Geoscientists estimate total mineral or fluid volumes within underground reservoirs. Medical imaging algorithms reconstruct 3D organ volumes from scan data using related integration techniques. Understanding how to set up and evaluate triple integrals is therefore a critical skill across STEM disciplines.
Worked example
Problem: Find the triple integral of f(x, y, z) = x · y · z over the box where x ∈ [0, 2], y ∈ [0, 3], z ∈ [0, 1].
Step 1 — Integrate over x: ∫[0 to 2] x dx = [x²/2] from 0 to 2 = (4/2) − 0 = 2. The inner result is 2 · y · z.
Step 2 — Integrate over y: ∫[0 to 3] 2y dy = 2 · [y²/2] from 0 to 3 = 2 · (9/2) = 9. The middle result is 9 · z.
Step 3 — Integrate over z: ∫[0 to 1] 9z dz = 9 · [z²/2] from 0 to 1 = 9 · (1/2) = 4.5.
Result: The triple integral equals 4.5. This can be verified by noting the function is separable: I = (∫[0,2] x dx)(∫[0,3] y dy)(∫[0,1] z dz) = (2)(9/2)(1/2) = 2 × 4.5 × 0.5 = 4.5. ✓
Physical interpretation: If x, y, z have units of meters and f = xyz represented a non-uniform density field in kg/m³, the integral value 4.5 would have units of kg·m³ (requiring dimensional checks with actual physical constants). In a pure mathematical context, the answer is simply the real number 4.5.
Limitations & notes
This calculator handles only rectangular (box-shaped) integration domains with constant limits. Non-rectangular regions — such as spheres, cylinders, tetrahedra, or regions bounded by curved surfaces — require variable limits that depend on the integration variables (e.g., x₁ = g(y, z)) and cannot be evaluated with this tool. For those cases, coordinate transformations to spherical or cylindrical coordinates, combined with the Jacobian, are typically employed. The supported integrand types are limited to six closed-form functions; arbitrary user-defined expressions (such as sin(xyz) or ln(x+y+z)) are not supported in this version due to the requirement for exact analytic antiderivatives. Numerical methods such as Gaussian quadrature or Monte Carlo integration should be used for integrands without clean antiderivatives. Additionally, this calculator assumes the function f is continuous and bounded on the domain, satisfying the conditions of Fubini's Theorem. Improper triple integrals over unbounded domains or with singularities require separate treatment using limits.
Frequently asked questions
What does a triple integral physically represent?
A triple integral represents the accumulated value of a three-dimensional function throughout a volume. When the integrand is 1, it gives the volume of the region. When the integrand represents density, it gives total mass. Other physical interpretations include total electric charge, gravitational potential, or probability over a 3D region depending on what f(x, y, z) models.
How do I set up the limits of integration for a triple integral?
For a rectangular box, limits are simply the constant bounds in each dimension: x from x₁ to x₂, y from y₁ to y₂, z from z₁ to z₂. For non-rectangular regions, you must express the inner limits as functions of the outer variables. For example, integrating over a tetrahedron or sphere requires expressing one variable's bounds in terms of the others, which is why choosing the correct order of integration and coordinate system (Cartesian, cylindrical, spherical) is critical.
What is Fubini's Theorem and why does it matter for triple integrals?
Fubini's Theorem states that if f(x, y, z) is continuous on a rectangular box, the triple integral equals the iterated integral and can be evaluated in any order (dx dy dz, dy dz dx, etc.) with the same result. This is crucial because it allows you to choose the most convenient order of integration — often determined by which order produces the simplest antiderivatives. Without Fubini's Theorem, the order of integration would need careful justification.
When should I use spherical or cylindrical coordinates instead of Cartesian?
Spherical coordinates (ρ, φ, θ) are ideal when the integration domain is a sphere, hemisphere, or cone, and when the integrand has the form f(x² + y² + z²). Cylindrical coordinates (r, θ, z) work best for cylinders, cones, and functions with radial symmetry in the xy-plane. Switching coordinates requires multiplying the integrand by the Jacobian determinant — r for cylindrical, ρ² sin(φ) for spherical — to correctly account for the volume element.
Can triple integrals be used to find the center of mass of a solid?
Yes. For a solid with density function ρ(x, y, z), the total mass is M = ∭ ρ dV, and the coordinates of the center of mass are x̄ = (1/M)∭ x·ρ dV, ȳ = (1/M)∭ y·ρ dV, z̄ = (1/M)∭ z·ρ dV. These four triple integrals over the solid's volume completely determine the center of mass, which is essential in structural engineering, aerospace design, and robotics.
Last updated: 2025-01-15 · Formula verified against primary sources.