Calculate Triple Integral with Variable Limits

Calculate the definite integral of the function $x$ from 0 to $\sqrt{xy}$ with respect to $z$, then from 0 to $x$ with respect to $y$, and finally from 0 to 1 with respect to $x$.

This problem revolves around the computation of a triple integral with variable limits, which is a classic multivariable calculus problem. Computing triple integrals can be particularly challenging due to the involvement of three variables and their respective limits of integration. The problem requires us to integrate sequentially over three variables: first, with respect to $z$, then $y$, and finally $x$. Each variable has specific limits which can be functions of the other variables, adding complexity to the problem.

A key concept in solving triple integrals is the understanding of how these variables and their limits interact. The inner integral is often dependent on the innermost variable, and its limits may be constants or functions of the other variables. Therefore, one important strategy is to make sure you evaluate the integrals in the correct order and respect the dependencies on the other variables. Be mindful of how each bound changes depending on the variable of integration, and don't forget that a triple integral represents a volume under a surface in a three-dimensional space.

Understanding this process enhances one’s ability to visualize and compute volumes in a multi-variable context, a fundamental skill in fields like physics and engineering. Familiarizing yourself with the strategies to tackle these integrals makes handling more complex systems manageable, especially when dealing with real-world applications where many interacting variables are the norm.