Triple Integral Calculator – Evaluate 3D Volume & Mass Integrals Online

This Triple Integral Calculator helps you evaluate integrals over 3D regions step-by-step. Enter your function, set variable limits, choose coordinate systems, and compute volume or mass easily — ideal for students and engineers learning multivariable calculus.

Last Updated: April 13, 2025

Triple Integral Calculator

Use this free online Triple Integral Calculator to evaluate integrals in three dimensions. Input your function, select coordinate systems, and compute the volume or mass of complex solids instantly with step-by-step explanations.

What Is a Triple Integral?

A triple integral extends the concept of single and double integrals to three-dimensional space. It's used to compute the volume under a surface in 3D, as well as physical quantities like mass, charge, or total energy when density is involved.

Formal Definition

A triple integral of a function \( f(x, y, z) \) over a region \( R \) is defined as:
\( \iiint_R f(x, y, z)\, dx\, dy\, dz \)

  • f(x, y, z): scalar field or density function
  • R: the 3D region of integration
  • dx dy dz: integration over the x, y, and z axes

Why It Matters

  • 📦 Compute volume of irregular 3D shapes
  • ⚖️ Calculate mass of solids with variable density
  • 🌡️ Model heat, pressure, and fluid flow in physics and engineering

Real-Life Analogy

Imagine trying to measure the amount of paint needed to fill an oddly shaped 3D object—too irregular for simple geometry. Triple integrals let you break it into tiny cubes, measure each, and sum them all to get an exact total.

How to Use the Triple Integral Calculator

  1. Enter the function \( f(x, y, z) \) to integrate
  2. Select your coordinate system: Cartesian, Cylindrical, or Spherical
  3. Define the bounds for x, y, and z (or r, θ, ϕ)
  4. Click "Calculate" to evaluate the triple integral
  5. Get the numerical result and optional step-by-step breakdown

Tip: Use this tool to validate homework problems or explore new coordinate systems in multivariable calculus.

Triple Integral Examples

  • Example 1: \( \iiint_R xyz\, dx\, dy\, dz \), over cube \( [0,1] \times [0,1] \times [0,1] \) → Result: \( \frac{1}{8} \)
  • Example 2: Volume under \( f(x, y, z) = 1 \) over a cylinder in cylindrical coordinates
  • Example 3: Mass of a solid with density \( \rho(x, y, z) = x^2 + y^2 + z^2 \) over a sphere

Applications of Triple Integrals

  • 🧱 Solid Volume Calculation
  • 🧲 Electromagnetic field modeling
  • 🧪 Thermodynamics and heat distribution
  • 🛠️ Mass, center of mass, and moment of inertia
  • 💧 Fluid dynamics and pressure computations

Frequently Asked Questions (FAQ)

What is a triple integral used for?

Triple integrals are used to compute volumes, mass, and other 3D physical quantities. They're essential in engineering, physics, and advanced calculus.

How do I set up a triple integral?

First define the bounds of your 3D region, then determine whether to use Cartesian, cylindrical, or spherical coordinates. Set up the integral using the limits and the function you're integrating.

What coordinate systems can I use?

Triple integrals can be evaluated in Cartesian (x, y, z), cylindrical (r, θ, z), or spherical (ρ, θ, φ) coordinates, depending on the symmetry of the region.

What does the result of a triple integral represent?

The result often represents the volume under a 3D surface, but can also represent total mass, charge, or other accumulated quantities when a density function is involved.

When should I switch coordinate systems?

If your region has circular, cylindrical, or spherical symmetry, switching to cylindrical or spherical coordinates can simplify the integration process significantly.

Start Exploring 3D Calculus Today

Use our Triple Integral Calculator to solve complex multivariable problems easily. Get accurate answers, explore different coordinate systems, and deepen your understanding of vector calculus.

Try the Calculator Now