Problem 122 - Calculus Volume 1

Question

For the following exercises, use shells to find the volumes of the given solids. Note that the rotated regions lie between the curve and the x-axis and are rotated around the y-axis. $$y=\frac{1}{x}, x=1, \text { and } x=100$$

Step-by-Step Solution

Answer

The volume of the solid is \(198\pi\) cubic units.

Step 1: Understand the Problem

We need to find the volume of the solid formed by rotating the region under the curve \( y = \frac{1}{x} \) between \( x = 1 \) and \( x = 100 \) around the y-axis. We'll use the method of cylindrical shells to find this volume.

Step 2: Set Up the Integral for Volume by Shells

Using the method of cylindrical shells, the volume is given by the integral: \[ V = \int_{a}^{b} 2\pi (radius)(height) \, dx \] In this case, \( radius = x \) and \( height = y = \frac{1}{x} \). Therefore, the integral becomes: \[ V = \int_{1}^{100} 2\pi x \cdot \frac{1}{x} \, dx = \int_{1}^{100} 2\pi \, dx \]

Step 3: Evaluate the Integral

Calculate the integral: \[ V = 2\pi \int_{1}^{100} 1 \, dx = 2\pi \left[x\right]_{1}^{100} = 2\pi (100 - 1) = 2\pi \times 99 \]

Step 4: Simplify and Find the Final Volume

Multiply to find the final volume: \[ V = 198\pi \] cubic units.

Key Concepts

Volume of Solids of Revolution

When we discuss the volume of solids of revolution, we're talking about volumes of 3D shapes formed by rotating a 2D area around an axis. Imagine the curve representing a region on a 2D plane. Rotating this region around a line, such as the x-axis, y-axis, or any other line generates a solid or 3D shape.
To find the volume of such solids, we often use techniques from calculus such as the disk method, washer method, or cylindrical shells method. Each method serves to break down the volume computation into manageable parts for integration.
For example, if you think about the method of cylindrical shells, you're essentially cutting the solid into thin "shells," each of which resembles a cylinder or tube. By adding up the volume of each shell, you get the volume of the entire solid.

Integral Calculus

Integral calculus is the heart of finding areas and volumes in higher mathematics. The basic idea is to divide a complicated shape into infinitely small pieces, and then sum up those pieces. This sum becomes the integral, which helps us find the area under a curve or the volume of a solid.
In the context of the cylindrical shells method, the integral calculus allows us to add up the volumes of an infinite number of cylindrical shells to find the total volume of the solid.
The integral like \[ V = \int_{a}^{b} 2\pi (radius)(height) \, dx \]is helpful to understand how each tiny piece contributes to the total volume. You integrate over the interval that includes all the x-values in your region of interest.

Rotating Regions

Rotating regions is a powerful concept in mathematics that allows us to transform a flat shape into a three-dimensional object. This process involves taking the area under a curve or between curves and spinning it around a line known as the axis of rotation.
With respect to the current problem, we're rotating the region under the curve \( y = \frac{1}{x} \)between \( x = 1 \) and \( x = 100 \) around the y-axis. This creates a solid whose volume can be challenging to compute with simpler methods, which is why we use cylindrical shells.
Understanding how to handle such rotational transformations is critical in advanced calculus, as it transcends just computing volumes to also understanding the nature and properties of the resulting solids.