Use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the x-axis.
y = 1/x

The x y-coordinate plane is given. A curve, a region, and two representative rectangles are graphed.
The curve starts at (1, 1), goes down and right becoming less steep, and ends at (7, 1⁄7).
The region is below the curve, above the x-axis, to the right of x = 1, and to the left of x = 7.
The region is divided into two parts by a dashed horizontal line at y = 1⁄7.
The first horizontal representative rectangle goes from (1, 0.67) to (1.5, 0.67).
The second horizontal representative rectangle goes from (1, 0.07) to (7, 0.07).

To solve the problem of finding the volume of the solid generated by revolving the region bounded by the curve \( y = \frac{1}{x} \), the x-axis, and the lines \( x = 1 \) and \( x = 7 \) about the x-axis, we will use the shell method.

### Shell Method Overview
The shell method involves integrating with respect to \( y \) when revolving around the x-axis. The general formula for the volume using the shell method is:

\[
V = 2\pi \int_{c}^{d} ( \text{radius} ) ( \text{height} ) \, dy
\]

### Problem Setup
1. **Radius**: When revolving around the x-axis, the radius of the shell is simply \( y \).
2. **Height**: The height of the shell at a given \( y \) is determined by the difference between the x-values on the curve. Given \( y = \frac{1}{x} \), solve for \( x \) to get \( x = \frac{1}{y} \). The height is therefore \( \left(\frac{1}{y} - 1\right) - \left(\frac{1}{y} - 7\right) = \frac{1}{y} - 1 - \frac{1}{y} + 7 = 7 - 1 = 6 \).

### Limits of Integration
The region is bounded vertically between \( y = \frac{1}{7} \) and \( y = 1 \). So the limits of integration are from \( y = \frac{1}{7} \) to \( y = 1 \).

### Volume Integral
\[
V = 2\pi \int_{\frac{1}{7}}^{1} y \cdot \left(\frac{1}{y} - 1 \right) \, dy
\]

Simplifying the integrand:

\[
V = 2\pi \int_{\frac{1}{7}}^{1} \left(1 - y\right) \, dy
\]

### Evaluate the Integral
Let's evaluate the integral:

\[
V = 2\pi \left[ \int_{\frac{1}{7}}^{1} 1 \, dy - \int_{\frac{1}{7}}^{1} y \, dy \right]
\]

The first integral is straightforward:

\[
\int_{\frac{1}{7}}^{1} 1 \, dy = y \Big|_{\frac{1}{7}}^{1} = 1 - \frac{1}{7} = \frac{6}{7}
\]

The second integral:

\[
\int_{\frac{1}{7}}^{1} y \, dy = \frac{y^2}{2} \Big|_{\frac{1}{7}}^{1} = \frac{1^2}{2} - \frac{\left(\frac{1}{7}\right)^2}{2} = \frac{1}{2} - \frac{1}{98} = \frac{49 - 1}{98} = \frac{48}{98} = \frac{24}{49}
\]

So the volume \( V \) becomes:

\[
V = 2\pi \left(\frac{6}{7} - \frac{24}{49}\right)
\]

Simplify the expression inside the parentheses:

\[
V = 2\pi \left(\frac{42}{49} - \frac{24}{49}\right) = 2\pi \cdot \frac{18}{49} = \frac{36\pi}{49}
\]

Thus, the volume of the solid is:

\[
V = \frac{36\pi}{49}
\]

Would you like a more detailed explanation, or do you have any questions? 

Here are some related questions:
1. How does the shell method differ from the disk/washer method?
2. What are some other examples of curves that can be revolved to create solids using the shell method?
3. How would the setup change if the region were revolved around the y-axis instead?
4. Can the shell method be used for regions that are not functions of x or y?
5. What are some limitations or advantages of the shell method compared to other methods?

**Tip**: When using the shell method, always check whether it's easier to integrate with respect to x or y, depending on the axis of revolution and the shape of the region.

Learn how to use the shell method in calculus to find the volume of the solid generated by revolving the region bounded by the curve y = 1/x, the x-axis, and lines x = 1 and x = 7 about the x-axis. This detailed problem solution covers the application of the shell method, integral calculus, and solving definite integrals. Suitable for college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation