Find the surface area of the volume generated when the following curve revolves around the y-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. (Round your answer to four decimal places.)
y = 
1

x
 from x = 1 to x = 5

The problem asks to find the surface area of the solid generated by revolving the curve \( y = \frac{1}{x} \) from \( x = 1 \) to \( x = 5 \) around the y-axis.

The formula for the surface area of a solid of revolution around the y-axis is given by:

\[
A = 2\pi \int_{x_1}^{x_2} x \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx
\]

Where:
- \( y = \frac{1}{x} \)
- \( \frac{dy}{dx} = -\frac{1}{x^2} \)
- The limits of integration are from \( x = 1 \) to \( x = 5 \).

### Step 1: Setting up the integral

Substitute into the formula:

\[
A = 2\pi \int_1^5 x \sqrt{1 + \left(-\frac{1}{x^2}\right)^2} \, dx
\]

First, simplify the expression inside the square root:

\[
1 + \left(\frac{dy}{dx}\right)^2 = 1 + \left(-\frac{1}{x^2}\right)^2 = 1 + \frac{1}{x^4}
\]

Thus, the integral becomes:

\[
A = 2\pi \int_1^5 x \sqrt{1 + \frac{1}{x^4}} \, dx
\]

### Step 2: Approximating the integral

Since this integral cannot be evaluated easily by hand, we will approximate it using a numerical method.

Let me calculate it for you.The approximate surface area of the solid generated by revolving the curve \( y = \frac{1}{x} \) from \( x = 1 \) to \( x = 5 \) around the y-axis is approximately **76.8030** square units (rounded to four decimal places).

Would you like more details on the process or have any further questions?

Here are five related questions to explore:
1. How would the surface area change if the curve revolved around the x-axis instead of the y-axis?
2. What if the curve was \( y = \frac{1}{x^2} \)? How would the surface area integral differ?
3. How would the surface area be affected if the limits were from \( x = 2 \) to \( x = 6 \)?
4. How can we compute the volume of this solid instead of its surface area?
5. What numerical methods can be used to approximate integrals like this one?

**Tip**: When setting up integrals for surface areas, always pay attention to whether the curve is revolving around the x-axis or the y-axis, as this changes the setup of the integral.

Find the surface area of the volume generated when the following curve revolves around the y-axis. If you cannot evaluate the integral exactly, use your calculator to approximate it. (Round your answer to four decimal places.) y = 1/x from x = 1 to x = 5.

Explore how to find the surface area of the solid generated when the curve y = 1/x is revolved around the y-axis from x = 1 to x = 5. This detailed solution includes key calculus concepts such as the surface area of revolution and numerical integration, making it suitable for college-level students.

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