Triple Integral Definition

Calculus 3 · Axiom Academy

One idea in three dimensions: chop a solid into tiny boxes, add up , and take the limit — then evaluate it as three nested integrals. 1. A Solid, Chopped Into Tiny Boxes Take a solid region E and slice it with planes into a grid of little boxes. A single box has volume . Pick a sample point inside each box, evaluate f there, multiply by , and add up every box. That sum is a Riemann sum in 3D. 2. Evaluating It: Three Integrals, One Inside the Next We never add up infinitely many boxes by hand — we integrate. If E sits between a bottom surface z=g_1(x,y) and a top surface z=g_2(x,y) over a base region D in the xy -plane, then the triple integral becomes an iterated integral : sweep z up the column first, then sweep that column over D . Hold x,y fixed and integrate z from the bottom surface g_1 up to the top surface g_2 — that's one vertical column. Slide the column across D in y , from the near edge to the far edge of the base region. Finally sweep in x across all of D . Three passes collapse the solid into a single number. Any of the six orders gives the same value — pick the one whose limits are easiest for the shape of E . 3. The Payoff: Volume and Mass Two choices of f make the triple integral pay off. With f=1 you accumulate pure volume. With , a density (mass per unit volume), you accumulate mass — each box contributes density its volume, exactly like a Riemann sum.

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