Triple Integrals in Rectangular Coordinates

Definition

The triple integral extends the concept of integration to functions of three variables over a solid region $D$ in three-dimensional space. $\iiint_D f(x, y, z) dV = \lim_{n \to \infty} \sum_{k=1}^n f(x_k, y_k, z_k) \Delta V_k$

• How to read: “Triple integral over D of f dV equals the limit as n goes to infinity of the sum from k equals one to n of f at (x-k, y-k, z-k) times delta V-k.”
• Meaning: The Riemann-sum definition: partition $D$ into small boxes, sample $f$ in each box, weight by box volume, and take the limit as boxes shrink.

Why It Matters

Triple integrals in Cartesian coordinates provide the most basic way to calculate the volume and mass of 3D objects. They are the universal starting point for spatial accumulation, essential for physics, fluid dynamics, and 3D engineering.

Core Concepts

• Volume Differential: In rectangular coordinates, $dV = dz dy dx$ (or any permutation).

• How to read: “dV equals dz dy dx.”
• Meaning: The infinitesimal volume element; order of $dz$, $dy$, $dx$ matches the order of nested integration limits.

• Iterated Integration: Evaluating the integral in three successive stages, usually finding the $z$-bounds (surfaces), then the $y$-bounds (curves), and finally the $x$-bounds (constants).
• Fubini’s Theorem (3D): The order of integration can be changed for continuous functions, though the limits must be adjusted accordingly.

Connected Concepts

• Triple Integrals in Cylindrical Coordinates
• Triple Integrals in Spherical Coordinates
• Moments of Inertia