The formula this page uses
∭_E f(x, y, z) dV = ∫ ∫ ∫ f dz dy dx cylindrical: dV = r dz dr dθ spherical: dV = ρ² sin φ dρ dφ dθ
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Triple Integral Calculator
Solve triple integral problems with clear steps, notation, and a final check.
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Evaluate the governing formula locally in your browser. Define each known quantity once, then change values to test another case instantly.
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Calculus
Triple Integral Calculator explained
The short version
- A triple integral adds something up through a solid: volume, mass, or average value.
- Three nested integrals, innermost first, and only the outermost pair of limits may be plain constants.
- Switching to cylindrical or spherical coordinates adds a factor, r or ρ² sin φ, that you must not forget.
What each part means
SymbolWhat it means
f(x, y, z) — The integrandDensity in kg per m³ for mass, temperature for an average, or 1 for plain volume.
E — The solid regionDescribed by three pairs of limits, innermost describing the deepest slice.
dV — Volume elementdz dy dx in rectangular, r dz dr dθ in cylindrical, ρ² sin φ dρ dφ dθ in spherical.
r, θ, z — Cylindrical coordinatesr is distance from the z-axis, θ is the angle around it, z is height. Good for pipes and cans.
Show your work: a full example
- The integral∫₀¹ ∫₀² ∫₀³ (x + y + z) dz dy dx
- Innermost in z, holding x and y still(x + y)z + z²/2 from z = 0 to z = 3
- Substitute3x + 3y + 4.5
- Middle integral in y3xy + 1.5y² + 4.5y from y = 0 to y = 2
- Substitute6x + 6 + 9 = 6x + 15
- Outer integral in x3x² + 15x from x = 0 to x = 1
- Substitute3 + 15 = 18
A second, different case
- A different case: a cylinder in cylindrical coordinatesradius 2, height 5, so r ≤ 2 and 0 ≤ z ≤ 5
- Set it up with the extra r∫₀^{2π} ∫₀² ∫₀⁵ r dz dr dθ
- Innermost in zr · 5 = 5r
- Middle in r∫₀² 5r dr = 5r²/2 from 0 to 2 = 5(4)/2 = 10
- Outer in θ∫₀^{2π} 10 dθ = 10(2π) = 20π
- As a decimal20π ≈ 62.83 cubic units
- Check against the school formulaπ r² h = π(2²)(5) = 20π, identical
Copy-ready example
∫₀¹ ∫₀² ∫₀³ (x + y + z) dz dy dx
The integral
Questions about the Triple Integral Calculator
When should I switch out of rectangular coordinates?
Look at the boundary. A round cross-section means cylindrical, because r ≤ 2 replaces the awkward −√(4 − x²) ≤ y ≤ √(4 − x²). A boundary at constant distance from the origin means spherical, because ρ ≤ 3 replaces x² + y² + z² ≤ 9.
What does a triple integral of 1 give me?
The volume of the solid, because you are adding up a value of 1 for every tiny box inside it. That is exactly the check used above, where ∭ r dz dr dθ reproduced π r² h.
Where does the ρ² sin φ come from in spherical coordinates?
A small spherical box has sides dρ, ρ dφ and ρ sin φ dθ. Multiplying those three lengths gives ρ² sin φ dρ dφ dθ. The sin φ shrinks the box near the poles, which is why the factor is there.
How do I find an average value instead of a total?
Divide the triple integral of f by the triple integral of 1 over the same solid. The second integral is just the volume, so average temperature equals total temperature-volume divided by volume.