Problem 42 - Thomas Calculus

Question

Find the volume of the solid generated by revolving about the \(x-\) axis the region in the first quadrant enclosed by the coordinate axes, the curve \(y=2 /\left(1+x^{2}\right),\) and the line \(x=1\).

Step-by-Step Solution

Answer

The volume is \(\frac{\pi^2}{8}\).

Step 1: Understand the Problem

We need to find the volume of a solid formed by revolving a region around the x-axis. The region is enclosed by the coordinate axes, the curve \(y = \frac{2}{1+x^2}\), and the line \(x = 1\).

Step 2: Set Up the Integral for the Volume

The volume of a solid of revolution generated by rotating a curve around the x-axis can be calculated using the disk method. The formula for the volume \(V\) is\[V = \pi \int_{a}^{b} [f(x)]^2 \, dx\]In this problem, \(f(x) = \frac{2}{1 + x^2}\), \(a = 0\) (the y-axis), and \(b = 1\) (the line \(x = 1\)).

Step 3: Write the Volume Integral

Substitute the function and the limits into the volume formula:\[V = \pi \int_{0}^{1} \left(\frac{2}{1+x^2}\right)^2 \, dx\]This simplifies to:\[V = \pi \int_{0}^{1} \frac{4}{(1+x^2)^2} \, dx\]

Step 4: Compute the Integral

To compute the integral \(\pi \int_{0}^{1} \frac{4}{(1+x^2)^2} \, dx\), we use the substitution \(x = \tan(\theta)\), \(dx = \sec^2(\theta) d\theta\). This converts the integral into:\[V = \pi \int_{0}^{\frac{\pi}{4}} 4\cos^2(\theta) \, d\theta\]Evaluate the trigonometric integral:\[V = 4\pi \int_{0}^{\frac{\pi}{4}} \frac{1 + \cos(2\theta)}{2} \, d\theta = 4\pi \left.\left( \frac{\theta}{2} + \frac{\sin(2\theta)}{4}\right)\right|_0^{\frac{\pi}{4}}\]After solving, you get:\(V = \pi \left[\frac{\pi}{4} + 0\right] = \frac{\pi^2}{8} \).

Step 5: Calculate the Result

Substitute back and simplify the expressions:\[ V = \frac{\pi^2}{8} \] The evaluation confirms the volume of the solid of revolution is computed correctly.

Key Concepts

Volume of Revolution

Calculating the volume of a solid of revolution involves imagining how a two-dimensional shape can form a three-dimensional object when rotated around an axis. By revolving a region or shape around the x-axis or y-axis, we form a volume that can be calculated through methods involving calculus.
This concept is particularly important in engineering and physical sciences, as it allows us to determine the capacity or space within a rotated shape, which could be anything from a bowl to a wine barrel.
The volume is essentially the sum of infinite thin disks or washers, each contributing to the total volume. By using calculus, we can set up an integral that accounts for all the infinitely small pieces, thus calculating the entire volume.

Disk Method

The disk method is a way of visualizing a solid of revolution. Imagine slicing the solid like a loaf of bread, where each slice is a disk. These disks have a thickness that approaches zero, and their radii are determined by the function being revolved.
Let's take our problem as an example: the function given is \( y = \frac{2}{1+x^2} \). When we rotate this curve around the x-axis, each disk perpendicular to the axis has a radius \( r = f(x) \), and thickness \( dx \).
The formula for the volume using the disk method is:

• \( V = \pi \int_{a}^{b} [f(x)]^2 \, dx \)

Where \( a \) and \( b \) are the bounds within which we revolve the shape. Each slice's volume \( \pi [f(x)]^2 dx \) is the area of the disk \(\pi r^2\), multiplied by its infinitesimal thickness \(dx\). This integral sums up all these tiny volumes.

Definite Integral

When solving calculus problems like finding volumes or areas, definite integrals come into play. The definite integral \( \int_{a}^{b} f(x) \, dx \) gives the net area between the curve \(y = f(x)\) and the x-axis, from point \(a\) to point \(b\).
In our exercise, the definite integral helps compute the volume of a solid by summing up the areas of disks over a specified interval. Here, the integral \( \pi \int_{0}^{1} \left(\frac{2}{1+x^2}\right)^2 \, dx \) calculates the total volume by narrowing the scope of integration to between the bounds 0 and 1, referenced by the problem stricture, like the line \(x=1\).
The purpose is to achieve precise calculations that add together the contributions of infinitely small sections of the solid. The definite integral, thus, transforms the abstract concept of volume into a calculated and tangible result.

Trigonometric Substitution

Trigonometric substitution is a strategy used to simplify integration, especially when dealing with complex rational functions or roots.
In many calculus problems, like the one we are examining, substitutions can help transform difficult integrals into more manageable forms.
For example, substituting \(x = \tan(\theta)\) simplifies the integral \( \pi \int_{0}^{1} \frac{4}{(1+x^2)^2} \, dx \) to the trigonometric equivalent:

• \( \pi \int_{0}^{\frac{\pi}{4}} 4\cos^2(\theta) \, d\theta \)

This substitution efficiently tackles the integral by transforming it into one that is solvable using trigonometric identities.
These techniques reveal the underlying patterns in integration and exploit trigonometric identities, such as \(\cos(2\theta)\), providing a way to solve integrals that initially appear difficult, by turning them into simpler expressions.