Question

Use the integration capabilities of a graphing utility to approximate the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. $$ y=\ln x, \quad y=0, \quad x=1, \quad x=3 $$

Short answer

The volume of the solid can be approximated by \(\pi \int_1^3 [\ln x]^2 dx\) where the integration is done numerically using a graphing calculator.

Step-by-step solution

Understand and Graph the Equations

First, sketch the graphs of \(y = \ln x\), \(y = 0\), \(x = 1\), and \(x = 3\). These will form the boundaries of the region we're interested in. The region is simply the area under the curve \(y = \ln x\) from \(x = 1\) to \(x = 3\), above the x-axis which is represented by \(y = 0\).

Apply Disk Method Formula

The general formula for the disk method is \[ V = \pi \int_a^b [R(x)]^2 dx \] where \(R(x)\) is the radius of a typical disc at \(x\) and \(a\) and \(b\) are, the endpoints of the interval.

Find the Radius

For the given problem, \(R(x)\) is simply \(y\), so it would be \(R(x) = \ln x\).

Substitute Values and Integrate

Putting all of this together, we get \[ V = \pi \int_1^3 [\ln x]^2 dx \] This is a problem for numerical approximation using a graphing utility.

Approximate the Integration using Graphing utility

Approximate the integral using a graphing calculator or other mathematical software capable of numerical integration to find the result.