# What is the triple integral?

Evaluates an iterated integral exactly where possible, checked numerically.

- Page: https://www.acalculator.org/math/triple-integral-calculator
- JSON spec: https://www.acalculator.org/math/triple-integral-calculator.json
- Version: 0b3d7ed55a15

## Default answer

Example with the default inputs (Function f(x, y, z) 5x - 3y, Order dz dy dx, Inner lower 0, Inner upper 1 - x - y, Middle lower 0, Middle upper 1 - x, Outer lower 0, Outer upper 1): The triple integral of 5x - 3y (dz dy dx) is 0.08333333333.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| f | Function f(x, y, z) | The integrand. |
| order | Order | dz dy dx: inner in z, then y, outer in x. |
| a1 | Inner lower | The inner lower limit. |
| b1 | Inner upper | The inner upper limit. |
| a2 | Middle lower | The middle lower limit. |
| b2 | Middle upper | The middle upper limit. |
| a3 | Outer lower | The outer lower limit. |
| b3 | Outer upper | The outer upper limit. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| exact | Exact value | The integral, exact. |
| value | Decimal value | The integral to 10 significant figures. |

## Method

Iterated CAS definite integrals, checked against tanh-sinh cubature.

## Assumptions

- Finite limits; radians. An answer that fails its check is not shown.

## Worked examples

1. f = 5x - 3y, order = dz dy dx, a1 = 0, b1 = 1 - x - y, a2 = 0, b2 = 1 - x, a3 = 0, b3 = 1 gives exact = 1/12, value = 0.083333. Source: OpenStax Calculus Vol. 3 (Strang and Herman, 2016), 5.4, Ex. 5.38.
2. f = 1, order = dx dy dz, a1 = -1 + z, b1 = 1 - z, a2 = -1 + z, b2 = 1 - z, a3 = 0, b3 = 1 gives exact = 4/3, value = 1.333333. Source: OpenStax Calculus Vol. 3 (Strang and Herman, 2016), 5.4, Ex. 5.39.

## FAQ

### What is a triple integral?

A triple integral adds up f(x, y, z) over a solid region in space. With f = 1 it is the volume of the solid; with f a density it is the mass. It is worked out as an iterated integral: integrate in the inner letter with the other two held constant, then in the middle letter, then in the outer letter.

### How do I enter the limits?

Type the order inner first, such as dz dy dx. The inner limits may use the two outer letters (here x and y), the middle limits may use the outer letter (x), and the outer limits must be numbers or constants. For the tetrahedron x, y, z ≥ 0, x + y + z ≤ 1, use z from 0 to 1 − x − y, y from 0 to 1 − x, and x from 0 to 1.

### How is the answer checked?

The exact value comes from a computer algebra system, one integral at a time. The page also works out the triple integral numerically, with a tanh-sinh rule in each letter, and the two must agree to 1 part in 100 million. If no exact form is found, or it fails the check, the page shows the numeric value alone, and only when two runs of the numeric rule with different step sizes agree.

### Why do I see only a decimal?

The algebra found no exact antiderivative for one of the integrals, its answer failed the check, or it took over 3 seconds. The decimal is the numeric value to 10 significant figures, shown only when two numeric runs agree to 1 part in 10 billion. Changing the order of integration sometimes gives an exact answer.

### Can I use cylindrical or spherical coordinates?

Yes: use your own letters in the order and put the Jacobian factor into the function. In cylindrical coordinates multiply by r (order dz dr dθ); in spherical coordinates multiply by ρ² sin φ, typed with plain letters such as p^2 sin(q). The volume of a ball of radius 1 is p^2 sin(q) with dp dq dt, p from 0 to 1, q from 0 to pi and t from 0 to 2pi, which is 4π/3.

### What is the difference from a double integral?

A double integral runs over a region of the plane and has two limits pairs; a triple integral runs over a solid and has three. For one-variable definite integrals use the integral calculator.

## Sources

- OpenStax, Calculus Volume 3, section 5.4 Triple Integrals: https://openstax.org/books/calculus-volume-3/pages/5-4-triple-integrals
- OpenStax, Calculus Volume 3, section 5.5 Triple Integrals in Cylindrical and Spherical Coordinates: https://openstax.org/books/calculus-volume-3/pages/5-5-triple-integrals-in-cylindrical-and-spherical-coordinates