Triple Integral Calculator
Free triple integral calculator for Cartesian, cylindrical and spherical coordinates. Enter the integrand and limits — inner limits may depend on the outer variables — and get the value with the Jacobian handled for you.
Evaluate ∭ f dV over any region you can describe with limits. Switch between Cartesian, cylindrical and spherical coordinates, use variable inner limits such as 1−x or √(1−x²), and let the calculator handle the Jacobian.
dV = dz dy dx
In terms of x, y, z
Limits of integration
Inner limits may use the outer variables — e.g. 1-x, sqrt(1-x^2)
How to evaluate a triple integral
- 1
Choose the coordinate system
Match the coordinates to the region: boxes suit Cartesian, cylinders and cones suit cylindrical, balls suit spherical.
- 2
Describe the region with limits
Work from the inside out, letting each inner limit depend on the variables outside it.
- 3
Enter the integrand
Type the function alone — the calculator supplies the Jacobian for cylindrical and spherical coordinates.
- 4
Read the value
The result appears with a closed form where one is recognised, and the full iterated integral is restated underneath.
The three coordinate systems
Cartesian
dV = dz dy dx
Best for boxes and regions bounded by planes or by graphs z = f(x, y).
Cylindrical
dV = r dz dr dθ
x = r cos θ, y = r sin θ, z = z. Best for cylinders, cones and paraboloids.
Spherical
dV = ρ² sin φ dρ dφ dθ
ρ is distance from the origin, φ is measured from the z-axis (0 to π), θ goes around (0 to 2π).
Worked triple integrals
What triple integrals compute
| Volume | ∭ 1 dV | Integrating the constant 1 measures the region itself. |
| Mass | ∭ ρ(x,y,z) dV | Density integrated over the solid. |
| Centre of mass | ∭ x·ρ dV ÷ mass | One integral per coordinate. |
| Moment of inertia | ∭ r²·ρ dV | r is the distance from the axis of rotation. |
| Average value | ∭ f dV ÷ volume | The mean of f over the region. |
| Charge | ∭ ρ_q dV | Charge density integrated over a volume, as in Gauss's law. |
Frequently asked questions
What is a triple integral? ▾
A triple integral adds up the values of a function over a three-dimensional region. Integrating the constant 1 gives the region's volume; integrating a density gives mass; integrating distance-squared times density gives a moment of inertia.
In what order are the limits applied? ▾
Inside out. The innermost integral is evaluated first and its limits may depend on both outer variables; the middle limits may depend on the outermost variable; and the outermost limits must be constants. If your outermost limits contain a variable, the setup is wrong.
When should I use cylindrical coordinates? ▾
When the region has an axis of rotational symmetry — cylinders, cones, paraboloids. Substituting x = r cos θ, y = r sin θ turns awkward circular boundaries into the simple limit 0 ≤ r ≤ R. Remember the volume element is r dz dr dθ.
When should I use spherical coordinates? ▾
When the region is a ball, a spherical shell, or a cone measured from the origin. The volume element is ρ² sin φ dρ dφ dθ, where ρ is the distance from the origin, φ is measured down from the positive z-axis, and θ goes around.
Why is there an extra r or ρ² sin φ? ▾
That is the Jacobian determinant of the coordinate change. A small box in r, θ, z space does not have the same volume as a box in x, y, z space, and the Jacobian is the correction factor. This calculator inserts it automatically, so enter only your integrand.
How does this calculator evaluate the integral? ▾
With nested 40-point Gauss–Legendre quadrature, which is exact for polynomials up to degree 79 in each direction and extremely accurate for smooth integrands. Recognisable answers such as π/3 or 4/3 are identified and shown alongside the decimal.
About this Calculator
Free triple integral calculator for Cartesian, cylindrical and spherical coordinates. Enter the integrand and limits — inner limits may depend on the outer variables — and get the value with the Jacobian handled for you.