Problem 31 - Essential Calculus

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. $$ y=2 \arctan (0.2 x), \quad y=0, \quad x=0, \quad x=5 $$

Step-by-Step Solution

Answer

The volume of the solid obtained by revolving the given region about the x-axis can be approximated using a graphing utility. The integral to be evaluated is \(V = \pi \int_0^5 [2 \arctan (0.2 x)]^2 dx\).

Step 1: Determine the Function

In this problem, the function to be revolved is given as \(y = 2 \arctan (0.2 x)\). The given bounds for \(x\) are from 0 to 5. As such, the region bounded by the function and the x-axis between these limits will be revolved to form the solid.

Step 2: Set Up the Integral for the Disk Method

The disk method requires setting up and evaluating an integral. The general formula for the volume of a solid of revolution using the disk method is \(V = \pi \int_a^b [f(x)]^2 dx\), where \(a\) and \(b\) are the bounds of the revolution (in this case 0 and 5), \(f(x)\) is the function being revolved, and \([f(x)]^2\) provides the radius of the disk at a given \(x\). Substituting these values into the formula gives us the integral \(V = \pi \int_0^5 [2 \arctan (0.2 x)]^2 dx\).

Step 3: Use a Graphing Utility to Approximate the Integral

With the integral set up, a graphing utility can be used to calculate its value. Calculators and computer software often have built-in functions to evaluate definite integrals. This can usually be done by inputting the function and bounds directly into the software and running the calculation. Make sure to enter \(f(x)\) as \(2 \arctan (0.2x)\) and the integral bounds as 0 to 5.

Key Concepts

Integration Techniques

Integration is a fundamental concept in calculus. It refers to the process of finding integrals, which can be thought of as the reverse operation of differentiation. In our problem, we use integration to calculate the volume of a solid formed by rotating a curve around an axis, specifically the x-axis.

Integrals can be classified as definite or indefinite. For our purposes, we are dealing with a definite integral, since we have specific limits of integration: from 0 to 5. This means we will be calculating the area under the curve of our function, between these two points.

• Definite integrals have upper and lower limits.
• The result of a definite integral is a numerical value.
• They can be approximated using numerical methods or graphing utilities.

Understanding how to set up the integral with the right function and limits is key to solving many calculus problems. It requires careful attention to the function that needs to be integrated, as well as the bounds of integration that define the area of interest.

Volume of Solids of Revolution

When we revolve a region around an axis, we generate a three-dimensional object known as a solid of revolution. The volume of this solid can be determined by calculus techniques such as integration.

The integral represents the cumulative sum of small disk-shaped slices that make up the volume of the solid. By calculating the integral, we effectively add up the volumes of these infinitesimally thin disks from the lower bound to the upper bound.

• The process is similar to adding up tiny pieces to form a whole.
• The axis of revolution, in our case the x-axis, plays a critical role in shaping the solid.
• This method is particularly useful for functions that are not easily described by simple geometric shapes.

Mastering this concept will help in solving many practical problems that deal with rotations and volumes in calculus.

Disk Method

The disk method is a specific integration technique used to find the volume of a solid of revolution. It's particularly effective when dealing with functions that are being rotated around the x-axis or y-axis.

In our problem, the disk method requires evaluating the expression \(\pi \int_a^b [f(x)]^2 dx\). This formula is derived from the formula for the volume of a cylinder, where \(R\) is the radius and \(h\) is the height, with volume being \(\pi R^2 h\). Each small disk within the integral has a radius given by the function value \(f(x)\) at that point.

• Each disk's volume is small, but together they form the entire solid's volume.
• The integration bounds (from 0 to 5) tell us where these disks begin and end along the x-axis.
• This method captures the idea of summing continuous slices to form a whole.

The disk method is a powerful example of how calculus can be used to solve real-world geometry problems involving rotation.

Graphing Utilities

Graphing utilities are powerful tools in calculus, helping with complex calculations that would otherwise be cumbersome by hand.

These utilities are often software applications or calculators that can graph functions, analyze curves, and most importantly, compute integrals. In the context of our problem, a graphing utility helps us approximate the integral we set up using the disk method.

• Enter the function into the graphing utility precisely as shown: \(2 \arctan (0.2x)\).
• Set the bounds from 0 to 5 to cover exactly the region intended.
• The utility then uses numerical integration techniques to find an approximate value of the integral.

By streamlining the integration process, graphing utilities make it faster and more efficient to solve problems, providing students with insight into function behavior and outcomes.