When solving triple integrals, the order of integration—whether to integrate with respect to $z$, $x$, or $y$ first—depends on the structure of the region of integration and the integrand's form. Choosing a particular order like first integrating with respect to $z$, then $x$, and finally $y$ might be advantageous for several reasons:

1. Region Simplicity:

Often, the region of integration is simpler to describe in one particular order. For instance, if the boundaries for $z$ depend only on $x$ and $y$, it is more convenient to integrate over $z$ first. This simplifies the limits of integration because you deal with constant or functionally simple bounds early.

2. Function Structure:

If the integrand (the function being integrated) is easier to handle when integrated with respect to $z$ first, this could be a strategic reason to choose that order. Sometimes, integrands separate into factors like $f(x, y)g(z)$, where integrating with respect to $z$ becomes straightforward and independent of $x$ and $y$, simplifying the calculations.

3. Variable Bounds:

The limits of integration may be simpler when integrating with respect to $z$, $x$, then $y$. For instance, when the boundaries of $z$ are functions of $x$ and $y$, you integrate over $z$ first because its bounds are well-defined in terms of these two variables.

4. Physical Interpretation:

In some applications, the $z$-direction may represent height or depth, and it could be natural to integrate over this variable first, especially when the region of interest is layered in the $z$-direction. This often occurs in volume calculations where the limits on $z$ vary based on the $x$-$y$-plane.

5. Reduction of Complexity:

Sometimes, by integrating with respect to $z$ first, you reduce the complexity of the remaining double integral over $x$ and $y$. This approach can simplify later steps by reducing the dimensions of the integral progressively.

Conclusion

The choice of the order depends on factors like the structure of the region of integration, function complexity, and limits of integration. Starting with $z$ often reflects the need to manage these issues in the most efficient way possible.

Would you like more details or further clarification?

Relative Questions:

1. How do different integration orders affect the final result of a triple integral?
2. What are examples of regions where integrating $z$-first simplifies calculations?
3. How do you handle changing the order of integration for complex regions?
4. In which applications (e.g., physics or engineering) does integrating $z$-first arise most naturally?
5. How can you determine the best order of integration for a given problem?

Tip:

When in doubt about the order of integration, sketching the region of integration can help visualize which variable should be integrated first for simplicity.