Triple Integral Calculator
How to compute a triple integral
Enter a polynomial in x, y and z (like 2xy + z) and constant limits for each variable. The calculator integrates term by term using ∫x^p dx = x^(p+1)/(p+1), then evaluates at the bounds and adds the results.
Worked example
∭ 1 dV over [0,1] × [0,2] × [0,3] = 1 × 2 × 3 = 6, the volume of the box. For ∭ x dV over the unit cube: (1²−0²)/2 × 1 × 1 = 0.5.
What is supported
Polynomial terms only (like 2x^2y, 3xy, z, 5) with constant rectangular bounds. Trigonometric, exponential, or variable-limit integrals are not supported yet, the calculator says so rather than guessing.
For a complete worked case, see the triple integral of xyz over a box solved one variable at a time.
Frequently asked questions
Over a box with f = 1 it gives volume. With a density function f(x,y,z) it gives total mass. In general it accumulates the function over a 3D region.
For continuous functions over a rectangular box, no, Fubini’s theorem says any order gives the same result.
This calculator handles polynomial terms exactly. Non-polynomial functions need different methods, so it refuses rather than returning a wrong answer.
How the three integrations nest
A triple integral over a box is three ordinary integrals nested inside each other. Work from the inside out. The innermost integration treats the other two variables as constants, and its result can still contain them. Integrate that result with respect to the next variable, then the last, until only a number remains.
Each pair of limits must stay matched to its differential: dx with the x limits, dy with the y limits, dz with the z limits. Swapping pairs without swapping limits gives a wrong number.
Example: Integrate f = x + 2z over x from 0 to 1, y from 0 to 2, z from 0 to 1
Step 1, Integrate with respect to x: The antiderivative is x^2/2 + 2zx. From 0 to 1, this gives 1/2 + 2z.
Step 2, Integrate with respect to y: Nothing in 1/2 + 2z contains y, so it acts as a constant: (1/2 + 2z) times y, from 0 to 2, gives 1 + 4z.
Step 3, Integrate with respect to z: The antiderivative of 1 + 4z is z + 2z^2. From 0 to 1, this gives 1 + 2 = 3.
Step 4, Read the result: The integral is 3. Size check: box volume 2, f from 0 to 3, so 0 to 6 is the plausible range.
When a triple integral is the right tool
Use one when a quantity is spread through a solid: total mass from a density that varies from point to point, total charge in a volume, or the average value inside a solid.
Common mistakes
- Mismatched limits: Applying z limits during the x step mixes variables and corrupts later steps.
- Discarding surviving variables: Terms in outer variables must stay. 2z survives the x step.
- Forgetting the lower limit: Each evaluation is upper minus lower, even when lower is 0.
- Skipping the size check: Compare with box volume times a typical value of f.
More questions
Compute the triple integral over the box, then divide by the box volume. For the worked example above, the average value is 3 divided by 2, which is 1.5.
This calculator covers rectangular boxes with constant limits. Curved regions need variable limits, and often cylindrical or spherical coordinates, which need a different setup.
A double integral accumulates over a flat, two dimensional region, giving area or volume under a surface. A triple integral adds a third dimension and accumulates over a solid region instead.