Problem 45 - Calculus and its applications

Question

Consider the function \(y=1 / x\) over the interval \([1, \infty)\) We showed in Section 5.3 that the area under the curve is not finite; that is, $$ \int_{1}^{\infty} \frac{1}{x} d x $$ diverges. Find the volume of the solid of revolution formed by rotating the area under the graph of \(y=1 / x\) over \([1, \infty)\) around the \(x\) -axis. That is, find \(\int_{1}^{\infty} \pi\left[\frac{1}{x}\right]^{2} d x\) This solid is sometimes referred to as Gabriel's horn.

Step-by-Step Solution

Answer

The volume of Gabriel's horn is \(\pi\).

Step 1: Set up the integral for volume

To find the volume of the solid of revolution, we use the formula of the volume of revolved areas. The formula is given as \( V = \int_{a}^{b} \pi \left[ f(x) \right]^{2} \, dx \) for a function \( f(x) \) over an interval \( [a, b] \). For Gabriel's horn, the function is \( f(x) = \frac{1}{x} \) and the interval is \( [1, \infty) \). Thus, the integral becomes \( \int_{1}^{\infty} \pi \left( \frac{1}{x} \right)^{2} \, dx \).

Step 2: Simplify the integral expression

First, simplify the expression inside the integral. Since \( \left( \frac{1}{x} \right)^{2} = \frac{1}{x^2} \), the integral becomes \( \pi \int_{1}^{\infty} \frac{1}{x^2} \, dx \). This simplification is crucial for evaluating the integral.

Step 3: Evaluate the improper integral

The integral \( \int_{1}^{\infty} \frac{1}{x^2} \, dx \) is an improper integral, so we evaluate it as a limit: \( \lim_{b \to \infty} \int_{1}^{b} \frac{1}{x^2} \, dx \).

Step 4: Integrate the function

Integrate \( \frac{1}{x^2} \) with respect to \( x \). The antiderivative of \( \frac{1}{x^2} \) is \( -\frac{1}{x} \). Therefore, the integral from 1 to \( b \) is \( -\frac{1}{b} + \frac{1}{1} \).

Step 5: Evaluate the limit

Now, find the limit as \( b \to \infty \) of \( -\frac{1}{b} + 1 \). As \( b \to \infty \), \(-\frac{1}{b}\) approaches 0. Thus, the expression becomes \( 1 - 0 = 1 \).

Step 6: Calculate the volume

Multiply the result from the integral step by \( \pi \) to find the volume. Since the integral evaluates to 1, the volume of the solid, Gabriel's horn, is \( \pi \cdot 1 = \pi \).

Key Concepts

Improper Integrals

Improper integrals are used in calculus to deal with scenarios where the interval of integration is infinite or the function has an asymptote within the range of integration. This is important because it allows us to evaluate integrals that otherwise wouldn't have a clear numerical result.

In the case of the function \(y = \frac{1}{x}\) over the interval \([1, \infty)\), we encounter an improper integral due to the upper limit being infinity. The integral in question is:

• \(\int_{1}^{\infty} \frac{1}{x} \, dx\) which diverges.

To properly compute such an integral, we take the limit as the upper bound approaches infinity:

• \(\lim_{b \to \infty} \int_{1}^{b} \frac{1}{x} \, dx\)

This method is crucial when evaluating certain physical and mathematical problems where infinite intervals or discontinuities are present. While the area under the curve for \(\int_{1}^{\infty} \frac{1}{x} \, dx\) diverges, this is not the case for the volume of revolution, as we'll see.

Volume of Revolution

The volume of revolution is a fascinating concept in calculus that allows us to find the volume of a three-dimensional object formed by rotating a two-dimensional area around an axis. For our case, we rotate the function \(y = \frac{1}{x}\) around the \(x\)-axis.

To calculate this, we use the formula:

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

For the interval \([1, \infty)\), the function \(f(x) = \frac{1}{x}\) simplifies the integral for the volume:

• \( \int_{1}^{\infty} \pi \left(\frac{1}{x}\right)^{2} \, dx \)
• This simplifies to \( \pi \int_{1}^{\infty} \frac{1}{x^2} \, dx \).

The simplification is key, as integrating \(\frac{1}{x^2}\) gives us a convergent result, unlike the original area integral. The approach evidences how modifying the function through squaring leads to convergence where previously it wasn't possible.

Gabriel's Horn

Gabriel's Horn, also known as Torricelli's Trumpet, is a famous geometric shape illustrating a captivating property of calculus. When the area under \(y = \frac{1}{x}\) on \([1, \infty)\) is revolved around the \(x\)-axis, it forms this infinite yet counter-intuitive figure.

Despite having an infinite surface area:

• The integral for the surface area \(\int_{1}^{\infty} 2\pi \left(\frac{1}{x}\right) \sqrt{1+\left(-\frac{1}{x^2}\right)^2} \, dx\) diverges.

The volume of Gabriel's Horn is finite:

• The integral \(\pi \int_{1}^{\infty} \frac{1}{x^2} \, dx\) converges to \(\pi\).

This paradoxical nature—a finite volume with an infinite surface area—demonstrates calculus's power to explore and explain the intricacies of infinite processes and boundaries. Gabriel's Horn reminds us of the surprising results calculus can produce and its essential role in understanding the geometry of infinite objects.