Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 11-20, sketch the solid \(S\). Then write an iterated integral for $$ \iiint_{S} f(x, y, z) d V $$ \(S\) is the region in the first octant bounded by the cylinder \(y^{2}+z^{2}=1\) and the planes \(x=1\) and \(x=4\).

Step-by-Step Solution

Answer

The iterated integral is \( \int_{1}^{4} \int_{0}^{1} \int_{0}^{\sqrt{1-y^2}} f(x, y, z) \, dz \, dy \, dx \).

Step 1: Understand the Region

The solid \(S\) lies in the first octant, so \(x, y, z \geq 0\). It is bounded by a cylinder \(y^2 + z^2 = 1\), which is a vertical cylinder with radius 1 around the x-axis. The planes \(x = 1\) and \(x = 4\) further bound the region, restricting the volume from \(x=1\) to \(x=4\).

Step 2: Identify Limits for x

Given the planes \(x=1\) and \(x=4\), the limits for \(x\) are straightforward. Therefore, \(1 \leq x \leq 4\).

Step 3: Identify Limits for y and z

Within the cylinder constraint \(y^2 + z^2 = 1\), the values of \(y\) and \(z\) are limited by \(-1 \leq y \leq 1\) and \(-1 \leq z \leq 1\), respectively, though the region only uses positive \(y\) and \(z\) in the first octant. Thus, \(0 \leq y \leq 1\) and \(0 \leq z \leq \sqrt{1 - y^2}\).

Step 4: Set Up the Iterated Integral

Compile the variable limits to set up the integral for the volume. First, integrate with respect to \(z\), then \(y\), and finally \(x\):\[ \int_{1}^{4} \int_{0}^{1} \int_{0}^{\sqrt{1-y^2}} f(x, y, z) \, dz \, dy \, dx \]

Key Concepts

Iterated Integrals

Iterated integrals are a powerful tool in multivariable calculus used to compute volumes of regions in three-dimensional space. When dealing with triple integrals, the process involves integrating a function of three variables successively across different axes. In this exercise, the goal is to integrate a function over the volume of a solid region, denoted by the region \(S\). This solid is situated in the first octant and is bound by certain geometric constraints such as a cylinder and planes.
The iterated integral is set up in stages, integrating first with respect to one variable while holding the others constant, and so on. In this example, we perform the integration in the order \(dz\), \(dy\), and finally \(dx\). This order is crucial because it reflects the bounds of integration for each respective variable, which are determined by the geometry of the solid. Setting up these integrals accurately is key to evaluating the volume correctly.
Iterating through the integrals enables precise computation of the volume or aggregate measure of properties over the defined region. This process can extend beyond simple volumes to evaluating more complex functions and their behavior over a region.

Cylindrical Coordinates

Cylindrical coordinates are often employed to simplify problems involving symmetrical structures, like cylinders. In our problem, the region is bounded by the cylinder equation \(y^2 + z^2 = 1\), aligning comfortably with cylindrical coordinates.
In cylindrical coordinates, a point in space is defined by three parameters: \(r\) for the radius or distance from the axis, \(\theta\) for the angle around the axis, and \(z\) for the height. For problems surrounding a cylinder, switching from Cartesian coordinates \((x, y, z)\) to cylindrical \((r, \theta, z)\) can reduce complexity and make the integration boundaries more natural.

• \(r\) corresponds to \(\sqrt{y^2 + z^2}\), which becomes 1 in this problem, due to the constraint imposed by the cylinder.
• \(\theta\) ranges from 0 to \(2\pi\), but as the region is in the first octant, it simplifies at 0 to \(\pi/2\).
• \(x\) acts as its own independent coordinate, since the boundaries are defined by the planes \(x = 1\) and \(x = 4\).

The approach of using cylindrical coordinates can greatly enhance calculations for three-dimensional shapes with rotational symmetry.

Integration Bounds

Integration bounds define the limits for each variable during an iterated integration process. In the context of our problem, these bounds are determined by the geometric constraints applicable to region \(S\).
The bounds for \(x\) are given directly by the planes \(x = 1\) and \(x = 4\). For the \(y\) and \(z\) axes, our constraints are defined by the circle's equation \(y^2 + z^2 = 1\) within the first octant. This restricts \(y\) and \(z\) as follows:

• \(y\) ranges from 0 to 1, ensuring that it remains within the cylinder.
• \(z\) is dependent on \(y\) and must satisfy \(0 \leq z \leq \sqrt{1-y^2}\), derived directly from rearranging \(z^2 = 1-y^2\).

These bounds ensure the integration process respects the physical dimensions of the region, and proper execution of these provides an accurate representation of the volume or function evaluation over the specified spatial domain.