Triple Integrals in Rectangular Coordinates

A triple integral calculates the total accumulation of a quantity over a three-dimensional region using a function of three variables. For example, if a function represents material density at each point in a solid metal block, the triple integral gives the block's total mass by accumulating density throughout the entire volume.

A triple Riemann sum approximates a triple integral by dividing a solid into small sub-boxes, estimating the function value at a sample point in each box, and multiplying by the small volume. As the sub-boxes become smaller, this sum approaches the exact triple integral value, providing the mathematical foundation for the integral's definition.

An iterated integral evaluates a triple integral by integrating one variable at a time in sequence. For instance, integrating with respect to z first accumulates values along the z-direction before integrating over the remaining variables. This step-by-step approach transforms the three-dimensional problem into manageable single-variable integrations.

Although Fubini's Theorem guarantees that any order of integration yields the same final result for continuous functions over bounded regions, different orders can simplify intermediate calculations. Some orders may involve easier antiderivatives or simpler limit expressions, making the computation more efficient without changing the accumulated quantity.

To find mass with variable density, set up a triple integral where the function is the density ρ(x, y, z) and integrate over the solid region. Each small volume element contributes mass equal to its density multiplied by its volume. Summing these contributions across the entire region yields the total mass.

For a rectangular box region, the triple integral has constant limits for each variable. The limits for x range from a to b, y from c to d, and z from e to h. These fixed bounds define the rectangular region and remain unchanged throughout the integration process regardless of the order chosen.

Partitioning divides the solid into small rectangular boxes, allowing the function to be approximated as constant within each box. Evaluating the function at a sample point in each box and multiplying by the box volume gives a local contribution. Summing all contributions creates a Riemann sum that converges to the triple integral as partitions become finer.