What is a triple integral?

A triple integral ∫∫∫_V f(x,y,z) dV integrates over a 3D region V. When f=1, it gives the volume. With f=ρ (density), it gives the mass of a solid.

When should I use cylindrical coordinates?

Use cylindrical coordinates for solids with circular symmetry around the z-axis: cylinders, cones, paraboloids. dV = r dr dθ dz with Jacobian r.

When should I use spherical coordinates?

Use spherical coordinates for spheres and cones symmetric about the origin. dV = ρ² sin(φ) dρ dφ dθ with Jacobian ρ²sin(φ). ρ is radial distance, φ is polar angle from z-axis, θ is azimuthal.

How accurate is the triple integral approximation?

The calculator uses 3D Simpson's rule with n=20 per dimension (20³ = 8000 evaluation points). For smooth functions this gives 4–6 significant figures of accuracy.

How do I compute the centroid of a 3D solid?

x̄ = ∫∫∫ x·ρ dV / ∫∫∫ ρ dV, and similarly for ȳ and z̄. Enter the density function in the Professional tab.

Triple Integral Calculator

Compute triple integrals over rectangular boxes in Cartesian, cylindrical, and spherical coordinates. Calculate volume, mass, and centroid of 3D solids.

Function f(x,y,z) ?

x lower

x upper

y lower

y upper

z lower

z upper

∫∫∫ f(x,y,z) dV over box (numerical approx.)

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Volume of box —

Average value of f —

Extended More scenarios, charts & detailed breakdown ▾

f(x,y,z)

x₁

x₂

y₁

y₂

z₁

z₂

∫∫∫ f dV Cartesian (numerical approx.)

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Professional Full parameters & maximum detail ▾

f(x,y,z)

x₁

x₂

y₁

y₂

z₁

z₂

Density ρ(x,y,z)

Triple Integral

∫∫∫ f dV (numerical approx.) —

Mass & Centroid

Mass = ∫∫∫ ρ dV (numerical approx.) —

Centroid x̄ —

Centroid ȳ —

Centroid z̄ —

How to Use This Calculator

Enter f(x,y,z) (e.g. x + y + z or 1 for volume).
Set the x, y, z bounds for the rectangular box.
Use the Cylindrical tab for rotationally symmetric solids (variables r, t, z).
Use the Spherical tab (variables rho, phi, theta) for spherical solids.
The Professional tab computes mass and centroid with a density function.

Formula

Cartesian: ∫∫∫ f dV | Cylindrical: ∫∫∫ f·r dr dθ dz | Spherical: ∫∫∫ f·ρ²sin(φ) dρ dφ dθ

Example

Volume of unit box: ∫₀¹∫₀¹∫₀¹ 1 dV = 1.000. Volume of sphere r≤1 in spherical: ∫₀¹∫₀^π∫₀^{2π} ρ²sin(φ) dρ dφ dθ = 4π/3 ≈ 4.189.

Frequently Asked Questions

A triple integral ∫∫∫_V f(x,y,z) dV integrates over a 3D region V. When f=1, it gives the volume. With f=ρ (density), it gives the mass of a solid.
Use cylindrical coordinates for solids with circular symmetry around the z-axis: cylinders, cones, paraboloids. dV = r dr dθ dz with Jacobian r.
Use spherical coordinates for spheres and cones symmetric about the origin. dV = ρ² sin(φ) dρ dφ dθ with Jacobian ρ²sin(φ). ρ is radial distance, φ is polar angle from z-axis, θ is azimuthal.
The calculator uses 3D Simpson's rule with n=20 per dimension (20³ = 8000 evaluation points). For smooth functions this gives 4–6 significant figures of accuracy.
x̄ = ∫∫∫ x·ρ dV / ∫∫∫ ρ dV, and similarly for ȳ and z̄. Enter the density function in the Professional tab.

Related Calculators

Sources & References (5) ▾

Triple Integrals — Paul's Online Math Notes — Lamar University
MIT OCW 18.02 Triple Integrals — MIT
Triple Integrals — Khan Academy — Khan Academy
Triple Integral — Wolfram MathWorld — Wolfram MathWorld
Marsden & Tromba — Vector Calculus (reference) — Macmillan