Problem 15 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 11-20, sketch the solid \(S\). Then write an iterated integral for $$ \iiint_{S} f(x, y, z) d V $$ $$ \begin{aligned} &S=\\{(x, y, z): 0 \leq x \leq 3 z \\ &\quad 0 \leq y \leq 4-x-2 z, 0 \leq z \leq 2\\} \end{aligned} $$

Step-by-Step Solution

Answer

The iterated integral is \( \int_{0}^{2} \int_{0}^{3z} \int_{0}^{4-x-2z} f(x, y, z) \ dy \ dx \ dz \).

Step 1: Analyze the Solid's Boundaries

The solid \(S\) is defined by the inequalities: \(0 \leq x \leq 3z\), \(0 \leq y \leq 4-x-2z\), and \(0 \leq z \leq 2\). These inequalities describe a region in \(xyz\)-space. \(x\) depends on \(z\), \(y\) depends on both \(x\) and \(z\), and \(z\) has a fixed range between 0 and 2.

Step 2: Visualize the Variable Dependencies

- For \(z\), consider a range from 0 to 2. - For each value of \(z\), \(x\) ranges from 0 to \(3z\). - For each \(x\), \(y\) ranges from 0 to \(4 - x - 2z\). This means as \(z\) increases, the possible values for \(x\) and \(y\) change, forming a tetrahedron-like shape in \(xyz\)-space.

Step 3: Set up the Iterated Integral

The iterated integral can be written by considering first \(y\), then \(x\), and finally \(z\). This is due to the dependency of \(y\) on \(x\) and \(z\), \(x\) on \(z\), and \(z\) being independent. \[\int_{0}^{2} \int_{0}^{3z} \int_{0}^{4-x-2z} f(x, y, z) \ dy \ dx \ dz\]

Step 4: Confirm Integration Order

Ensure the integration order matches the given constraints. We integrate \(y\) from 0 to \(4-x-2z\), then \(x\) from 0 to \(3z\), and finally \(z\) from 0 to 2.

Key Concepts

Multivariable Calculus

In multivariable calculus, we explore functions that have more than one variable. This expands on the concept of ordinary calculus where functions involve a single independent variable. Multivariable calculus is essential for understanding phenomena in multiple dimensions, like in physics and engineering.

It deals with vector functions and scalar functions of several variables. For example, a function like \( f(x, y, z) \) involves three variables, typically representing dimensions in space. Multivariable calculus concepts such as partial derivatives and multiple integrals provide tools to analyze and compute over these functions.

• Partial derivatives help us understand the function’s behavior with respect to each independent variable.
• Multiple integrals, like double and triple integrals, allow us to compute areas, volumes, and more over a specified region.

This makes multivariable calculus vital for modeling and solving complex real-world problems.

Solid Regions in Space

When dealing with multiple integrals in multivariable calculus, we often focus on solid regions in space. These regions are described by inequalities involving the variables. In the given problem, the solid \(S\) is defined by the conditions:

• \(0 \leq x \leq 3z\)
• \(0 \leq y \leq 4-x-2z\)
• \(0 \leq z \leq 2\)

These inequalities carve out a specific part of the 3D space, usually forming a distinct geometric shape, like a wedge or a tetrahedron.

Visualizing these regions can be crucial for setting up integrals correctly. By diagramming the dependencies and ranges, you develop a clearer sense of how each variable behaves within the solid. Understanding the region allows for more intuitive and accurate integration, as you can better foresee the changes in boundaries and limits when switching integration orders.

Integration Order

Choosing the correct integration order is a key aspect of solving an iterated integral problem. The integration order is determined based on the dependencies between variables. Each variable corresponds to one level of integration, from innermost to outermost.

The integration order affects both the complexity of solving the integral and the ease of evaluating the bounds. In the solution:

• \(y\) is integrated first from 0 to \(4-x-2z\) because it depends on both \(x\) and \(z\).
• \(x\) is integrated next from 0 to \(3z\), showing its dependency only on \(z\).
• Finally, \(z\) is independent and integrated from 0 to 2.

By organizing the integration in this order, we respect the natural dependencies of the regions’ boundaries as seen in the inequalities, minimizing the risk of miscalculations.

Coordinate Dependencies

Understanding coordinate dependencies is crucial for effectively solving iterated integrals. Dependencies indicate how one variable influences another in the defined region.

In the exercise, notice how each variable relies on certain conditions set by the others:

• \(x\) is bound by \(z\), from 0 to \(3z\), meaning that \(x\)'s possible values are limited by \(z\)'s current value.
• \(y\)'s upper limit changes based on both \(x\) and \(z\), making it more complex to evaluate independently.

The different dependencies create a layered structure of constraints, with each layer influencing the next. This layered understanding ensures accurate determination of the region's shape and size when solving the integral.

Developing this practical cognizance of how coordinates affect each other allows you to better set up and solve integrals, ensuring all dependencies are respected and correctly represented in your solutions.