Evaluating a Triple Integral with Variable Limits

Evaluate the integral $\displaystyle \int_{0}^{3} \int_{0}^{x} \int_{0}^{x-y} 4xy \, dz \, dy \, dx$.

In this problem, we are tasked with evaluating a triple integral with the limits defined by variables. This type of integral can be challenging as it requires a good understanding of not just integration, but also how to interpret the geometric region over which we're integrating. For triple integrals, the integration is performed over a three-dimensional region, which in this case is bounded by the planes given by the limits of integration. We work from the innermost integral to the outermost one, carefully tracking the interaction between the variables as specified by the given bounds.

The order of integration is important in triple integrals and may be chosen strategically to simplify computation based on the symmetry or other features of the integrand and the limits of integration. In this problem, determining the bounds involves examining the relationships between the variables x, y, and z. Since the limits involve one variable depending on another, it's crucial to understand how each variable interacts within the specified region.

It's helpful to visualize the integration region, possibly sketching the shapes or using technology to model them. This visualization aids in understanding the flow of the integration and the changes necessary when handling the bounded region step by step. Such problems often advance a student's facility with analytic skills and illustrate the application of multi-variable calculus principles.