Triple Integrals: Find Volume and Mass in 3D Easily

Double integrals accumulate over flat regions. But the real world is three-dimensional: a solid block of metal has a density that varies with x, y, and z, and a cloud of gas fills a volume. To add up a quantity spread through a solid — its mass, its electric charge, or simply the size of the region itself — you need one more layer of integration.

A triple integral, written ∫E f(x, y, z) dV, sums the values of f over every tiny box dV inside a solid region E. When f = 1, the integral gives the volume of E; when f is a density function, it gives the total mass. You compute it as a three-times-iterated integral: integrate in z first, then y, then x — working from the inside out and treating the outer variables as constants at each step.

Key Concepts

• dV = dz dy dx (in any order) — a tiny box of volume.
• Iterated form: integrate the innermost variable first; its limits may depend on the outer variables.
• Projection method: for a solid under z = g₂(x, y) and above z = g₁(x, y), the inner limits are g₁ to g₂ and the outer double integral runs over the shadow region in the xy-plane.
• Cylindrical: dV = r dz dr dθ; spherical: dV = ρ² sin φ dρ dθ dφ.
• Mass = ∫ (density) dV; center of mass and moments of inertia extend naturally from the 2D case.

Formulas

• ∫E f dV = ∫ab ∫h₁(x)h₂(x) ∫g₁(x,y)g₂(x,y) f(x, y, z) dz dy dx
• Volume of E = ∫E 1 dV.
• Average value of f over E: (1 / volume(E)) · ∫E f dV.

Worked Example

Find the volume of the solid under z = 6 − x − y and above the rectangle 0 ≤ x ≤ 1, 0 ≤ y ≤ 2 in the xy-plane.

1. Set up. Volume = ∫01 ∫02 ∫06−x−y 1 dz dy dx. (The height 6 − x − y stays positive since x ≤ 1, y ≤ 2.)
2. Inner integral: ∫06−x−y dz = 6 − x − y.
3. Middle integral (y from 0 to 2): ∫02 (6 − x − y) dy = [(6 − x)y − y²/2]02 = 2(6 − x) − 2 = 10 − 2x.
4. Outer integral (x from 0 to 1): ∫01 (10 − 2x) dx = [10x − x²]01 = 10 − 1 = 9.
5. Answer: 9 cubic units. Rough check: the height averages about 6 − 0.5 − 1 = 4.5 over a base of area 2, and 4.5 × 2 = 9. It matches.

Common Misconceptions

• “The innermost integral must always be dz.” Any order works — choose the order that makes the limits simplest for your solid.
• “Inner limits can depend on anything.” Inner limits may involve the outer variables, but never a variable that has already been integrated away.
• “Triple integrals only give volumes.” With f = 1 you get volume; with a density you get mass; with other choices you get charge, total heat, or average temperature.

Real-World Applications

• Mechanics: mass and center of mass of 3D objects with non-uniform density.
• Physics: total charge in a volume from a charge density; gravitational field of extended bodies.
• Engineering: fluid mass in a tank, heat content stored in a solid part.
• Probability: probabilities over 3D regions from joint density functions.

A triple integral is three single integrals nested: work inside out, let inner limits depend on the outer variables, and pick the coordinate system that matches the solid’s symmetry — cylindrical for tubes, spherical for balls. With f = 1 you measure pure volume; with a density function you weigh the solid.