Triple Integrals in Rectangular Coordinates

Triple integrals provide a method for calculating the accumulated value of a function over a three-dimensional region. Common applications include computing volume, mass, and other physical quantities that vary with position. The fundamental idea is to partition a solid region into small rectangular boxes, evaluate the function at sample points within each box, and sum the contributions. As the partitions become finer, this triple Riemann sum approaches the exact value of the triple integral.

In rectangular coordinates, a triple integral is typically expressed as:

\begin{equation*}\iiint_B f(x,y,z)\,dV = \int_a^b \int_c^d \int_e^h f(x,y,z)\,dz\,dy\,dx\end{equation*}

or with another order of integration, depending on how the accumulation is organized. The integral is evaluated iteratively as a sequence of single-variable integrations. For example, integrating with respect to z first accumulates values along the z-direction before integrating over the remaining variables. Other orders, such as integrating with respect to x or y first, represent different ways of sweeping through the three-dimensional region.

For a continuous function on a rectangular box, Fubini’s Theorem states that the order of integration can be changed without changing the value of the integral. This flexibility is useful because some orders of integration may be easier to evaluate than others. Although the final accumulated quantity remains the same, the choice of order can simplify the intermediate calculations.

In practical applications, such as calculating the mass of a metal block with variable density ρ(x, y, z), the triple integral

\begin{equation*}M=\iiint_B \rho(x,y,z)\,dV\end{equation*}

represents the block's total mass. Each small volume element contributes an amount of mass equal to its density multiplied by its volume. Summing these contributions over the entire region gives the total mass.