Multiple integrals

The formula

Single integral, for reference — accumulate along one axis:

I = ∫ f(x) dx        (over an interval a ≤ x ≤ b)

Reading: sum f(x) times a sliver of width dx across the interval.

Double integral — accumulate over a flat region R in the plane:

I = ∬_R f(x,y) dA      dA = dx dy

Reading: chop the region into tiny tiles of area dA, multiply each tile by the value of f sitting on it, add them all up.

Triple integral — accumulate over a solid region V in space:

I = ∭_V f(x,y,z) dV    dV = dx dy dz

Reading: same idea, but now the pieces are little bricks of volume dV filling a solid.

Iterated form (how you actually compute it) — Fubini's theorem lets you do it one axis at a time:

∬_R f(x,y) dx dy = ∫_c^d [ ∫_a^b f(x,y) dx ] dy

Reading: integrate over x first with y held fixed, then integrate that result over y. The inside integral collapses one dimension; the outside integral finishes the job.

The mass-property trio, all of which are just weighted multiple integrals:

Mass                 m   = ∭_V ρ dV
Center of mass (CG)  x̄   = (1/m) ∭_V x·ρ dV        (and ȳ, z̄ the same way)
Inertia              I_z = ∭_V (x² + y²)·ρ dV       (about the z-axis)

Reading: mass is density summed over the solid; the center of mass is the mass-weighted average position (for uniform density this coincides with the geometric centroid; for a real multi-material part it does not); the moment of inertia weights each bit of mass by the square of its distance from the axis.

Where you meet it

Mass-properties review on a flight component. Somebody asks for the CG and the inertia tensor of a bracket or a payload before it goes on the shaker or the spin table. Those numbers are triple integrals of density over the part's geometry. The CAD package runs them for you, but when the model and the measured CG disagree, you're the one who has to know what the software was actually computing.
Beam and section work at the desk. The area moment of inertia I = ∬ y² dA and the section modulus that set a beam's stiffness and bending stress are double integrals over the cross-section. For a rectangle it collapses to b·h³/12; for a welded-up or machined profile you're integrating the real shape.
Antenna and aperture power on the RF bench. Total power radiated or captured by an aperture is the field intensity integrated over the aperture area — a double integral over the dish or the array face. Directivity and effective area fall out of the same surface integrals.
Fluid load on a tank or gate. The total force and the center of pressure on a submerged wall are pressure integrated over the wetted area. Get the center of pressure wrong and your anchor bolts see the wrong moment.

How it works

A multiple integral is one number that answers "how much, total, over this whole region?" The trick that makes it computable is Fubini's theorem: a double integral equals a nested pair of ordinary single integrals, done one axis at a time. Integrate over x with y frozen, get a function of y, then integrate that over y. A triple integral peels the same way, one dimension per pass.

The order usually doesn't matter — but "usually" is doing real work in that sentence. Fubini's theorem lets you swap the order of integration only when the function is absolutely integrable over the region, meaning ∬|f| dA is finite. When it isn't, swapping the order can hand you two different answers, both wrong. Tonelli's companion result covers the case where f never goes negative: for a non-negative function you may always swap, because the pieces can only pile up, never cancel. In bench-and-CAD engineering work — density, pressure, power, all non-negative — you're almost always safe. The trap lives in signed integrands and improper regions that run to infinity.

The limits are where the real mistakes happen. For a rectangular region the bounds are constants and the setup is bookkeeping. For anything else — a triangle, a wedge, a fillet, a pie slice — the inner limits depend on the outer variable, and you have to describe the region's edge as a function before you write a single bound. Swapping integration order means re-deriving those limits from scratch; you cannot just move the dx and dy and copy the numbers over. That copy-the-numbers move is the single most common blown answer.

The second big lever is picking coordinates that match the region. A circle, a cylinder, or a sphere is miserable in x-y-z and clean in polar, cylindrical, or spherical. But you can't just substitute — every change of variables drags a Jacobian factor with it. In polar, dA = r dr dθ, not dr dθ. Drop that r and your area comes out wrong by a factor that grows with radius. In spherical, dV = r²·sinφ dr dφ dθ, where φ is measured down from the z-axis (the polar angle) and θ is the azimuth — and note that many physics and engineering texts swap the names θ and φ, so check the convention before copying a formula; putting the sine on the azimuth is exactly the Jacobian blunder this paragraph is warning about. Forgetting the Jacobian is the number-two blown answer, right behind bad limits.

A few more things that bite. Units ride along with every differential — a triple integral of density (mass per volume) times dV returns mass, cleanly, but only if you tracked the volume element's units. Moment of inertia depends entirely on which axis you picked; I about the centroid and I about an edge differ by the parallel-axis term, and a mass-properties table that doesn't state the reference point is useless. And when the geometry has no closed form — a real casting, a measured pressure field — the integral becomes a numerical sum over a mesh, which is exactly what a finite-element solver is doing under the hood. The math doesn't change; the tiles just get counted by a computer.

History

The idea of slicing a region into infinitely many thin pieces and adding them up is older than calculus itself. Bonaventura Cavalieri, a professor at Bologna, laid it out in his 1635 book Geometria indivisibilibus continuorum — geometry by "indivisibles" ^[1]^[2]. His principle: two solids of the same height with equal cross-sectional areas at every level have the same volume. That's a volume integral stated in words, a full half-century before Newton and Leibniz. Cavalieri had the method in manuscript by the late 1620s — MacTutor puts the completed text in December 1627 ^[1] — but sat on it for six years out of deference to Galileo, who was planning something similar ^[4]. His Swiss contemporary Paul Guldin attacked the indivisibles as logically shaky — and he wasn't entirely wrong; the rigor came later ^[1].

For the next two and a half centuries, engineers and mathematicians computed double and triple integrals by treating them as repeated single integrals and trusting that the order of integration didn't matter. It usually didn't, so nobody lost sleep. But "usually" isn't a theorem, and the counterexamples — functions where switching the order gives two different answers — showed the trust needed a foundation.

That foundation arrived with the Italian analyst Guido Fubini, who in 1907 proved the general theorem that now carries his name: if a function is absolutely integrable over a region, the multiple integral equals the iterated single integrals, in either order ^[3]. Two years later, in 1909, Leonida Tonelli published the companion result for non-negative functions, which is the version working engineers lean on without knowing its name — because density, pressure, and power don't go negative ^[3]. Between Cavalieri's slices and Fubini's theorem sits the whole story of how a good physical intuition finally got the rigor to back it up.

Related tools

/tools/moment-of-inertia-shapes
/tools/section-modulus-rect
/tools/beam-deflection
/tools/antenna-gain-beamwidth
/tools/convert-area-volume
/tools/tank-volume-horizontal
/tools/tank-volume-vertical
/tools/concrete-slab-volume
/tools/concrete-column-volume

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →