Triple Integral Calculator
Triple integrals compute volume, mass and flux over three-dimensional regions — the kind of problem where a Cartesian region like a box has straightforward bounds but the solid between two paraboloids requires careful order-of-integration decisions. This calculator evaluates ∭f(x,y,z) dV over the bounds you specify, supports Cartesian, cylindrical and spherical coordinates, and shows each antiderivative step.
How to compute a triple integral
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1
Enter f(x,y,z)
The integrand. Standard notation: x*y*z, x^2+y^2, sin(x)*cos(y).
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2
Choose a coordinate system
Cartesian (dx dy dz), cylindrical (r dr dθ dz), or spherical (ρ² sin(φ) dρ dφ dθ).
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3
Set the bounds
For each of the three variables — constants or functions of the others.
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4
Pick integration order
dzdydx, dxdydz, etc. Choice can dramatically simplify the maths.
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5
See step-by-step evaluation
Inner integral first, then middle, then outer, with antiderivatives at each stage.
What the three coordinate systems are for
| System | Volume element | Best for |
|---|---|---|
| Cartesian | dx dy dz | Boxes, prisms, general non-symmetric regions |
| Cylindrical | r dr dθ dz | Cylinders, cones, surfaces of revolution |
| Spherical | ρ² sin(φ) dρ dφ dθ | Balls, sectors of spheres, gravitational problems |
Using the wrong system turns a trivial integral into a nightmare. A ball of radius 1 integrated in Cartesian has messy √(1 − x² − y²) bounds; in spherical, it’s ∫₀²π ∫₀π ∫₀¹ ρ² sin(φ) dρ dφ dθ, clean and separable.
Common problems
- Mass: ∭ρ(x,y,z) dV, where ρ is density.
- Centre of mass: ∭x ρ dV / total mass, analogous for y and z.
- Moment of inertia: ∭r² ρ dV about a chosen axis.
- Volume: ∭1 dV — the integrand is 1, reducing to computing the region’s volume.
Changing the order of integration
For a region where the inner bound can’t be expressed nicely as a function of the outer variable, swapping order often helps. Sketch the region, project onto the inner-outer plane you want, and re-derive bounds.
Worked example: volume of a sphere
In spherical coordinates, the unit ball {x²+y²+z² ≤ 1}:
V = ∫₀²π ∫₀π ∫₀¹ ρ² sin(φ) dρ dφ dθ
= ∫₀²π ∫₀π [ρ³/3]₀¹ sin(φ) dφ dθ
= ∫₀²π ∫₀π (1/3) sin(φ) dφ dθ
= ∫₀²π (1/3)[-cos(φ)]₀π dθ
= ∫₀²π (2/3) dθ
= 4π/3
The famous V = (4/3)πr³ drops out in three clean steps — in Cartesian the same integral is multiple pages.
Numerical fallback
Some integrals have no closed-form antiderivative. When symbolic integration fails, the calculator falls back to numerical quadrature, returning an approximate value with an error estimate.
Frequently Asked Questions
Most often the bounds were wrong. Triple-integral bounds can depend on inner variables, and mis-ordering produces mathematically different integrals. Sketch the region first, then derive bounds carefully.
The calculator switches to numerical methods (adaptive quadrature). You get a numerical answer with an error bound rather than a symbolic expression.
Spherical when the region has full 3D symmetry about a point (balls, cones from a point). Cylindrical when there’s axial symmetry (cylinders, surfaces of revolution around an axis). Cartesian when there’s neither.
No. All computation runs in your browser.
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