Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.
y=ln x,y=ln x^2; and y=ln 8; about the y-axis

Verified step by step guidance

1. First, identify the region R bounded by the curves: \(y = \ln x\), \(y = \ln x^2\), and \(y = \ln 8\). Rewrite \(y = \ln x^2\) as \(y = 2 \ln x\) to better understand the boundaries.
2. Next, express the boundaries in terms of \(x\) for a given \(y\). From \(y = \ln x\), solve for \(x\) to get \(x = e^y\). From \(y = 2 \ln x\), rewrite as \(y = \ln x^2\) and solve for \(x\) to get \(x = e^{y/2}\). The horizontal boundary is \(y = \ln 8\).
3. Determine the interval for \(y\) over which the region exists. Since \(y\) ranges from the lower intersection point of the curves up to \(y = \ln 8\), find the intersection points by setting \(\ln x = 2 \ln x\), which simplifies to \(\ln x = 0\), so \(x=1\), and find corresponding \(y\) values.
4. Set up the volume integral using the shell method since the region is revolved about the y-axis. The radius of a shell at position \(x\) is \(x\), and the height is the vertical distance between the curves in terms of \(x\). The height is \(\ln 8 - \max(\ln x, 2 \ln x)\) depending on the region.
5. Write the volume integral as \(V = 2 \pi \int_{x=a}^{x=b} (\text{radius})(\text{height}) \, dx\), where \(a\) and \(b\) are the \(x\)-values corresponding to the intersection points and the boundary \(y=\ln 8\). Evaluate the integral to find the volume.

Key Concepts

Logarithmic Functions and Their Graphs

Understanding the behavior and properties of logarithmic functions like y = ln x and y = ln x^2 is essential. These functions define the boundaries of the region R, and knowing their graphs helps visualize the area to be revolved. For example, y = ln x^2 can be rewritten as y = 2 ln x, which affects the shape of the region.

Volume of Solids of Revolution

This concept involves finding the volume generated when a region is revolved around an axis. Methods such as the shell method or the disk/washer method are used depending on the axis of rotation. Since the region is revolved about the y-axis, the shell method is often convenient for volumes involving functions of x.

Setting Up and Evaluating Definite Integrals

To find the volume, one must set up an integral with proper limits corresponding to the intersection points of the curves. This involves expressing variables appropriately (e.g., x in terms of y) and integrating the volume element over the interval. Accurate evaluation of this integral yields the volume of the solid.