Use this Triple Integral Calculator to enter values, adjust options, and review results in a compact responsive workspace.
Results are calculated automatically as you enter data.
Enter an integrand and coordinate bounds.
Cartesian uses x, y, z bounds. Cylindrical uses r, theta, z and applies Jacobian r. Spherical uses rho, phi, theta and applies Jacobian rho^2 sin(phi).
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A triple integral is a way to add up values throughout a three-dimensional region. Instead of adding values along a line or across a flat area, it accumulates a function through a volume.
The basic idea is:
Here, \(E\) is the solid region, \(f(x,y,z)\) is the quantity being measured at each point, and \(dV\) represents a tiny piece of volume. If \(f(x,y,z)=1\), the triple integral gives the volume of the region. If \(f\) represents density, temperature, charge density, or another quantity that varies through space, the triple integral gives a volume-weighted accumulation of that quantity.
Triple integrals are especially useful because many real objects and mathematical regions are three-dimensional. They let you calculate volumes, masses, total charge, average values, and other quantities that depend on position in space.
Triple integrals connect geometry with accumulation. They are used when a quantity is spread throughout a solid region rather than concentrated at one point.
For example:
Even when an exact symbolic integral is difficult or impossible to find, a numerical method can still estimate the value closely enough for exploration, checking homework, or building intuition.
A triple integral starts by breaking a solid region into many small boxes or cells. The function is sampled inside each cell, multiplied by the cell’s volume, and then all contributions are added together. As the cells become smaller, the sum approaches the exact integral when the function and region satisfy the usual conditions for integration.
For a rectangular box in Cartesian coordinates, a triple integral can be written as:
The exact order of \(dx\), \(dy\), and \(dz\) can change for rectangular boxes without changing the value, as long as the limits match the same region.
This calculator uses adaptive cubature. In every coordinate cell it evaluates embedded tensor-product 3-point and 5-point Gauss-Legendre rules. Their difference estimates the local numerical error.
The local error indicator is \(|Q_5(C)-Q_3(C)|\). The calculator first splits the full region into eight cells, then repeatedly divides the cell with the largest estimated error into eight smaller cells. Refinement stops only when the combined estimated error meets the requested relative tolerance or a safety limit is reached.
Where:
The coordinate system determines how each quadrature point is converted into Cartesian \(x\), \(y\), and \(z\) before the integrand is evaluated.
| Coordinate mode | Coordinate variables | Cartesian conversion | Jacobian factor |
|---|---|---|---|
| Cartesian | \((x,y,z)\) | \((x,y,z)=(x,y,z)\) | \(1\) |
| Cylindrical | \((r,\theta,z)\) | \(x=r\cos\theta,\ y=r\sin\theta,\ z=z\) | \(r\) |
| Spherical | \((\rho,\phi,\theta)\) | \(x=\rho\sin\phi\cos\theta,\ y=\rho\sin\phi\sin\theta,\ z=\rho\cos\phi\) | \(\rho^2\sin\phi\) |
The Jacobian matters because equal steps in cylindrical or spherical coordinates do not always represent equal physical volumes. For example, a small angular step farther from the origin covers more space than the same angular step near the origin.
If the integrand is \(1\), the triple integral gives volume. For a box with:
The volume is:
So the triple integral equals \(6\) cubic coordinate units.
A cylinder of radius \(2\) and height \(3\) is naturally described with cylindrical coordinates:
Using \(f(x,y,z)=1\), the volume integral is:
The factor \(r\) is the cylindrical Jacobian. Evaluating the integral gives:
This matches the familiar cylinder formula \(\pi r^2h\).
A sphere of radius \(2\) centered at the origin is naturally described with spherical coordinates:
For volume, use \(f(x,y,z)=1\):
The spherical Jacobian is \(\rho^2\sin\phi\). The exact volume is:
This example also shows why angle units matter: \(\pi\) and \(2\pi\) are radian measures, not degree measures.
The main result is an approximation of the triple integral over the selected coordinate bounds. Its meaning depends on the integrand.
If the integrand is \(1\), the result represents volume. If the integrand is a density, the result represents a total amount such as mass. If the integrand is temperature or another field quantity, the result is a volume-weighted total, and the average value is often easier to interpret.
The average value is:
Where \(V(E)\) is the region volume. The average value has the same units as the integrand. For example, if \(f\) is measured in kilograms per cubic meter and the coordinates are measured in meters, the integral is in kilograms while the average value remains in kilograms per cubic meter.
The displayed region volume is based on the same coordinate weights used in the integral. In Cartesian coordinates, volume is usually the product of side lengths for a rectangular box. In cylindrical and spherical coordinates, the Jacobian changes the volume, so the volume is not usually just the product of the coordinate-widths.
The preview value range shows the smallest and largest integrand values found on a separate bounded preview grid. It is helpful for spotting negative values, unusually large values, or unexpected variation, but it is not guaranteed to be the true global minimum or maximum of the function and does not drive the error estimate.
Use a triple integral when you need to accumulate a quantity over a three-dimensional region.
Common use cases include:
Cartesian coordinates are often best for boxes and rectangular solids. Cylindrical coordinates are often best for cylinders, tubes, cones, and regions with circular symmetry around an axis. Spherical coordinates are often best for spheres, balls, shells, and regions centered around the origin.
A numerical triple integral is an approximation, not a proof of the exact value. A smaller relative tolerance requests tighter agreement between embedded quadrature rules and can require substantially more function evaluations. The displayed estimated error is a numerical indicator, not a rigorous proof for every possible function.
This calculator uses constant rectangular boxes in the selected coordinate variables. It does not handle variable-dependent or infinite limits. Improper, singular, discontinuous, or highly oscillatory integrals may fail to converge before the depth or evaluation limit; in that case the result is clearly marked as an unconfirmed estimate.
Standard coordinate domains are enforced: \(r\) and \(\rho\) must be nonnegative, spherical \(\phi\) must stay between \(0\) and \(\pi\), and an angular interval cannot be wider than \(2\pi\). These restrictions prevent negative Jacobian weights and accidental multiple coverage of the same physical region.
Displayed values are rounded for readability; nonzero values outside the ordinary display range use scientific notation, while only exact signed zero is shown as zero. The rounded display should not be treated as an exact guarantee. For homework, engineering, physics, or other high-stakes uses, verify the setup and compare with exact methods, a more specialized numerical method, or guidance from a qualified instructor or professional.
Supported expressions include variables \(x\), \(y\), and \(z\); constants such as \(\pi\) and \(e\); arithmetic operators; parentheses; and common functions such as sin, cos, sqrt, abs, exp, ln, log, min, and max.
A triple integral calculates a volume-weighted accumulation over a three-dimensional region. If the function is \(1\), the result is volume. If the function represents density or another spatial quantity, the result is the total amount of that quantity over the region.
The Jacobian corrects for how coordinate cells stretch or shrink when mapped into ordinary Cartesian space. In cylindrical coordinates, the volume factor is \(r\). In spherical coordinates, the volume factor is \(\rho^2\sin\phi\).
No. Enter the integrand using \(x\), \(y\), and \(z\). In cylindrical or spherical mode, sample points are converted into Cartesian coordinates first, and then the integrand is evaluated at those \(x\), \(y\), and \(z\) values.
Angles are entered in radians. For a full circle, use \(2\pi\). For a half turn, use \(\pi\).
No. A smaller tolerance asks the adaptive method to refine until its error indicator is smaller, but it does not make a numerical estimate exact. Singularities, discontinuities, sharp peaks, or oscillations can remain unreliable; check that the outcome says converged and compare important results independently.
In Cartesian coordinates, a rectangular box volume is the product of the side lengths. In cylindrical and spherical coordinates, coordinate widths must be weighted by the Jacobian, so the physical volume usually differs from the simple product of the coordinate intervals.
No. This style of calculator uses constant lower and upper bounds for each coordinate direction. Bounds such as \(z\) from \(0\) to \(x+y\) or \(r\) from \(0\) to \(2\sin\theta\) require a more general iterated-integral setup.
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