Triple Integral of a Linear Function Over a Rectangular Region

Integrate the function $x + y + z$ from 0 to 1 with respect to $x$, then from 0 to 2 with respect to $z$, and finally from 0 to 3 with respect to $y$.

In this problem, we are asked to evaluate a triple integral of the linear function x plus y plus z over a rectangular region in three-dimensional space. This is a typical problem in multivariable calculus where we explore the concept of integrating functions over regions in 3D space. The integral is performed in a specific order: first with respect to x, then z, and finally with respect to y. This order of integration is essential for correctly evaluating the triple integral, and understanding the setup is key. When tackling such problems, the region of integration is crucial. Here, the rectangular region is defined by very straightforward limits for x, y, and z. These limits form a 3D box where each variable is independent of the others within its range. This independence simplifies the integration process as you're essentially finding the volume under the surface described by the function within this box.

Conceptually, this problem allows students to practice and understand how triple integrals can be used to calculate volumes in higher dimensions or other related physical quantities like mass or charge distribution if the function represents a density. Furthermore, this problem provides insight into the application of Fubini's Theorem, which allows the evaluation of multiple integrals as an iterated process of single-variable integrals, given the regions of integration are rectangular or can be decomposed as such. This foundational understanding will aid in further studies where more complex regions or functions are involved.