Question

List the six possible orders of integration for the triple integral over the solid region \(Q\) \(\iint_{Q} \int x y z d V\). $$ Q=\left\\{(x, y, z): 0 \leq x \leq 2, x^{2} \leq y \leq 4,0 \leq z \leq 2-x\right\\} $$

Short answer

The valid orders of integration for this problem are: (x, z, y), (z, x, y), (z, y, x), (y, z, x). The orders (x, y, z) and (y, x, z) are not applicable due to their limits.

Step-by-step solution

Enumerate possible orders

The six possible orders can be remembered by recalling the six permutations of (x, y, z). The possible orders of variables in the triple integral are: 1. (x, y, z) 2. (x, z, y) 3. (y, x, z) 4. (y, z, x) 5. (z, x, y) 6. (z, y, x) Please note that the ranges for each variable must match the correct order of integration, which means the boundaries for the integral have to be adjusted accordingly.

Check validity of orders

Not all of the orders can be applied to the given integral. In order for an order to be valid, the limits for each variable must be constants or functions of the other variables. For the given problem, the limits for (x, y, z) and (y, x, z) are not in this form. Therefore, these orders are not applicable. The valid orders are: (x, z, y), (z, x, y), (z, y, x), (y, z, x).

Key concepts

Multiple Integrals

When delving into the world of calculus, one significant concept encountered is that of multiple integrals. These are an extension of regular single-variable integrals into the realm of multiple variables. Specifically, in the case of a triple integral, we are integrating a function of three variables over a three-dimensional region.

Imagine you have a scalar function, such as density or temperature, within a solid shape, and you want to find a cumulative value, like total mass or energy. A triple integral allows you to compute such a value by integrating over the volume of the solid. Visually, the process can be imagined as adding up the function's values at infinitely small cubic sections throughout the entire volume.

Each integral in the series—from a double to triple integral—takes into account an additional dimension, accounting for more complexity and providing a more descriptive representation of physical phenomena.

Integration Limits

The integration limits in multiple integrals are fundamental in defining the region over which the integration will occur. These limits are not always constants; they can be functions defining boundaries that vary according to the other variables involved.

In the context of a triple integral, the limits for the outermost integral are typically constants, representing the bounds along one axis. The limits for the next integral can be functions of the outer variable, and similarly, the innermost integral can have limits as functions of the two outer variables.

One key to solving multiple integrals correctly is setting these limits appropriately. Incorrect limits can lead to erroneous results, as they would imply integration over an unintended region. Each permutation of the variables for the order of integration will have a corresponding, and sometimes unique, set of limits tailored to describe the solid region accurately.

Calculus

At the heart of these problems lies calculus, the mathematical discipline focused on change and motion. While beginning calculus students often work with functions of a single variable, advanced calculus includes those involving multiple variables, leading to concepts like multiple integrals.

Calculus provides the tools for analyzing and understanding the geometry and physics of multi-dimensional spaces. It helps in tackling real-world problems where quantities are not constant but vary across different dimensions. Whether it is calculating the trajectory of a satellite or predicting the flow of heat in a rod, calculus is the language that describes these changes.

The triple integral problem is an example of how integrals are generalized in higher dimensions. With each added dimension, calculus expands the horizons of what can be modeled and solved, exemplifying the complexities and beauties of mathematical exploration.

Solid Region Integration

Understanding the concept of solid region integration is critical in handling triple integrals efficiently. When we perform integration over a solid region, we are interested in finding an aggregate quantity across the entire volume of the solid. The solid region is defined by bounds on all three axes in a three-dimensional space.

In practical terms, to perform a triple integral over a solid region, one must first comprehend the shape and limitations of this region, which is dictated by the integration limits. These limits are described by inequalities, as seen in the exercise, where the solid region is constrained by certain conditions on each of the variables x, y, and z.

The correct approach to solid region integration requires careful consideration of these bounds, and the order in which you integrate—and consequently, the limits you use—can simplify the process. Translating physical or geometric constraints into mathematical expressions is a key step in solving integrals over solid regions, enabling the application of calculus to derive meaningful answers from complex three-dimensional scenarios.