Free Online
Triple Integral Calculator
Solve triple integrals for volume, mass, and physics applications. Accurate, fast, and easy to use.
Triple Integral Calculator
Enter your function using the keypad or keyboard.
Result will appear here
Master 3D Integration
Triple integrals allow you to work in three dimensions. Our tool helps you break them down.
3D Volumes
Calculate the volume of complex 3D regions defined by inequalities.
Physics Ready
Ideal for calculating mass, center of mass, and moments of inertia.
Step-by-Step
Understand the process of iterated integration in 3 variables.
What is a Triple Integral?
A triple integral is an integral of a function of three variables, f(x, y, z), over a region in three-dimensional space. It is denoted by three integral signs: ∫∫∫.
While a double integral calculates volume under a surface, a triple integral can be thought of as calculating the "hypervolume" of a 4D object, or more commonly, the total mass of a 3D object with variable density. If the function f(x, y, z) = 1, the triple integral calculates the volume of the region itself.
Coordinate Systems
Choosing the right coordinate system is crucial for solving triple integrals efficiently.
Cartesian
(x, y, z)
dV = dx dy dzCylindrical
(r, θ, z)
dV = r dr dθ dzSpherical
(ρ, θ, φ)
dV = ρ²sin(φ) dρ dθ dφHow to Use This Calculator for Triple Integrals
Solve triple integrals by iterating through the variables one by one:
- 1Start with the innermost integral. Enter the function and integrate with respect to the first variable (e.g., dz).
- 2Take the result and integrate it with respect to the second variable (e.g., dy).
- 3Finally, integrate that result with respect to the outermost variable (e.g., dx).
Frequently Asked Questions
What is the physical meaning of a triple integral?
It often represents the total mass of an object if the function represents density. If the function is 1, it represents the volume of the region.
How do I choose the order of integration?
The order (e.g., dz dy dx) depends on the geometry of the region. Choose the order that makes the bounds of integration simplest to define.
Can this calculator handle spherical coordinates?
You can manually convert your function to spherical coordinates and then use the calculator to integrate the resulting expression.
Is it free?
Yes, our triple integral solver is completely free.