Evaluate Triple Integral of Polynomial Function

Evaluate the triple integral $\int_{-2}^{3} \int_{0}^{2} \int_{0}^{1} x^3 y z^2 \, dz \, dx \, dy$.

Evaluating a triple integral involves understanding the concept of integration in three dimensions. This type of problem is typically encountered in multivariable calculus where functions of several variables are integrated over a three-dimensional region. In this case, the integral bounds define a rectangular box in 3D space, over which the function is integrated. This requires performing integration iteratively for each variable, starting from the innermost integral. Given the nature of integration, it exploits the concept of the Fundamental Theorem of Calculus while extending it to multiple dimensions.

Key to solving triple integrals is the setup of the integral bounds and understanding the order of integration. When dealing with a triple integral, recognize that the order of integration - dz, dx, dy - dictates the sequence in which you integrate each variable. Each iteration reduces the dimensionality of the region. Thus, after integrating with respect to z, you end up with a double integral over x and y, and so forth. The function $x^3 y z^2$ is a straightforward polynomial which makes the integration process more direct, as the antiderivatives follow standard power rules.

The process also involves evaluating the given limits correctly and requires meticulous attention to detail to keep track of each step. The complexity of triple integrals scales with the intricacy of the function and region of integration, and it's essential to visualize the 3D region over which you are integrating. Since this problem is set within straightforward limits and involves a polynomial, it is an excellent practice to grasp core techniques in triple integration without the added complexity of changing coordinate systems or irregular boundaries.