Question

List the six possible orders of integration for the triple integral over the solid region \(Q\) \(\iint_{Q} \int x y z d V\). $$ Q=\left\\{(x, y, z): 0 \leq x \leq 1, y \leq 1-x^{2}, 0 \leq z \leq 6\right\\} $$

Short answer

The six possible orders of integration for the triple integral over the given solid region Q are: dy dz dx, dy dx dz, dz dx dy, dz dy dx, dx dy dz, dx dz dy.

Step-by-step solution

Recognize the Solid Region Q

Take a good look at the region Q. Its boundaries are defined by the equations \(0 \leq x \leq 1\), \(y \leq 1 - x^{2}\), and \(0 \leq z \leq 6\). Since \(y\) is dependent on \(x\), and \(z\) is independent of both \(x\) and \(y\), these can be considered as indicators for setting up the orders of integration.

List possible orders of integration

Based on the interdependencies of the variables, the six possible orders of integration are: - dy dz dx - dy dx dz- dz dx dy- dz dy dx - dx dy dz - dx dz dy

Key concepts

Solid Region Integration

Understanding solid region integration in multivariable calculus is crucial for solving volume-related problems. The process involves calculating the integral over a three-dimensional region, often referred to as a 'solid'. The boundaries of this solid are typically defined by functions of the variables involved — in this case, they are described by equations involving coordinates \(x, y, z\).
In our exercise, the solid region, denoted \(Q\), is bound by \(0 \leq x \leq 1\), \(y \leq 1 - x^{2}\), and \(0 \leq z \leq 6\). These bounds must be carefully considered when setting up the integral to ensure that all points within the region are accounted for. It's much like creating a custom-fit box around an object; every side must be the right size to fully encompass it without leaving gaps or extending beyond its dimensions.
While identifying the boundaries, we observe that \(z\)'s range is constant and independent of \(x\) and \(y\), whereas \(y\)'s range is directly affected by the value of \(x\). This hierarchical relationship of variables suggests a strategy for determining the order of integration. A clear understanding of these relationships is essential in selecting the correct order to simplify the integration process as much as possible.

Multivariable Calculus

At the heart of solid region integration lies multivariable calculus. This branch of mathematics focuses on functions of several variables and includes the study of limits, derivatives, and integrals. When dealing with multiple variables, the complexity increases because the functions can change in more than one direction, and our integration strategy must account for this multi-dimensional nature.
In our specific problem, we are asked to integrate a function with respect to a volume \(dV\), which requires the computation of a triple integral. The function we're integrating, \(xyz\), depends on all three variables. Therefore, we must consider how each variable interacts with the others within the given region \(Q\). This is where the orders of integration become particularly important — each different order can potentially simplify the integration process by taking advantage of the bounds and relationships between the variables.

Volume Integration

Volume integration is the process of finding the volume of a solid region by integrating over its entire extent. This is analogous to adding up infinitesimally small volumes (akin to tiny blocks) that fill up the entire region. When presented with a triple integral such as \(\iint_{Q} \tint xyz \, dV\), volume integration is used to determine the combined product of functions across the volume of \(Q\).
To approach the integral, one typically considers the most natural and simplest order of integration, based on the region's bounds. The six possible orders listed in the step-by-step solution demonstrate the different sequences by which we can perform integrations along \(x\), \(y\), and \(z\). Each sequence might lead to different intermediate steps, and choosing the most convenient sequence can greatly simplify the calculation. Some orders might result in less complex integrals due to the way the limits of integration naturally unfold, which makes certain choices preferable depending on the context of the problem.