Triple Integral Calculator
Approximate three-dimensional integrals for volume, mass, charge, and probability using constant or nested bounds.
A triple integral calculator evaluates ∭f(x, y, z) dV over a three-dimensional region. It computes volume, mass, charge, and average value by integrating with respect to x, y, and z in sequence. Accurate results require the correct integrand, variable order, and integration bounds for the specified region.
Use constant lower and upper limits for x, y, and z.
Best for rectangular boxes or any region where each variable has fixed bounds such as x: 0 to 1, y: 0 to 2, z: 0 to 3.
Numerical approximation of the triple integral
Region Measure
0Approximate integral of 1 over the region
Average Value
0Integral divided by region measure
Sample Count
0Midpoint evaluations used
Integration Summary
This tool estimates the integral numerically. For sharper accuracy on curved regions or rapidly changing functions, increase the subdivisions per axis.
How to Use the Triple Integral Calculator
A triple integral accumulates a quantity throughout a three-dimensional region. This calculator evaluates the integral in the order dz dy dx using a midpoint approximation, so it is useful for fast estimates when you know the bounds and integrand.
Enter the Integrand
Type the function of x, y, and z that you want to integrate, such as x^2 + y*z or exp(-(x+y+z)).
Choose Bounds
Use fixed bounds for rectangular regions or switch to nested bounds when the middle or inner limits depend on the outer variables, such as y = 0 to 1 - x or z = 0 to x + y.
Adjust Accuracy
Increase subdivisions if your region has curved boundaries or your function changes quickly. Lower values are faster, while higher values generally improve the estimate.
Common Setup Reference
Use these patterns to translate standard triple-integral problems into calculator inputs.
| Region Type | Example Bounds |
|---|---|
| Rectangular Box | x: 0 to a, y: 0 to b, z: 0 to c |
| Triangular Prism | x: 0 to 1, y: 0 to 1-x, z: 0 to h |
| Tetrahedron | x: 0 to 1, y: 0 to 1-x, z: 0 to 1-x-y |
| Variable Ceiling | x: 0 to 2, y: 0 to 1, z: 0 to x+y |
| Decay Model | f: exp(-(x+y+z)) on a box |
Real Example Use Cases
Mass from Variable Density
If density varies throughout a solid, use the density function as the integrand. The calculator estimates the total mass by summing that density across the entire three-dimensional region.
Accumulated Heat or Charge
When temperature or charge density changes in space, a triple integral gives the total heat content or total charge within the region. This is common in engineering and electromagnetics.
Three-Variable Probability
If the integrand is a probability density function, integrating over a bounded region estimates the probability mass concentrated in that three-dimensional subset of the sample space.
Integral Facts & Notes
3 Variables
A triple integral accumulates a quantity across x, y, and z over a three-dimensional region.
dz dy dx
This calculator evaluates the integral in a fixed nested order from z to y to x.
n^3
Using n subdivisions per axis produces n cubed midpoint samples in the approximation.
Numerical
The result is an estimate, not a symbolic closed form, so accuracy depends on resolution and smoothness.
Frequently asked questions (FAQ)
Does this triple integral calculator find an exact answer or a numerical result?
This calculator is a numerical solver. It uses midpoint-rule integration to estimate the triple integral, so the result is an approximation rather than an exact symbolic antiderivative or closed-form answer.
What function or equation syntax can I use in the integrand?
You can enter a function or equation-style expression such as x^2 + y*z, sin(x), exp(-(x+y+z)), or sqrt(x+y+1). The integrand can use the variables x, y, and z, along with constants like pi and e.
How do variable bounds and limits work for the integration region?
The outer x bounds set the main domain first. Then the y limits can depend on x, and the z bounds can depend on both x and y, which lets the calculator model a nested region with variable limits.
Can I use this calculator with different coordinate systems?
This version works in a Cartesian coordinate system using cartesian coordinates x, y, and z. It does not directly accept cylindrical coordinates or spherical coordinates, so those problems need to be rewritten before using the solver.
Why is my result negative?
A triple integral can be negative when the function is negative over much of the region, or when the limits are reversed in one variable. Check both the sign of the integrand and the order of the bounds before recalculating.
How many subdivisions should I use for more accurate integration?
Start around 16 to 24 for smooth functions over a simple domain. Increase the subdivisions when the graph of the function changes quickly or the region has sloped boundaries, keeping in mind that higher settings take longer to compute.
Can I use this calculator for volume, mass, or probability problems?
Yes. If the integrand is 1, the triple integral gives the volume of the selected region. If the function represents density or a probability model, the result gives the accumulated total over that domain.
Does this solver draw a graph of the domain or region?
No. This calculator focuses on the numerical result and integration setup, not on rendering a graph. If you need visualization, use the reported bounds and limits in a separate graphing tool.
What happens if the function or equation is invalid?
The calculator shows an error when the function, equation, or bounds cannot be parsed, or when evaluation produces a non-finite result. This usually means the integrand or one of the variable limits needs to be corrected.