https://file.aiquickdraw.com/mathbot/file/1726240012269thn35im0.jpg

The problem is asking to find the volume of a solid generated by revolving a region around the line \(x = 4\). The region is bounded by the following curves:

- \(xy = 2 \implies x = \frac{2}{y}\)
- \(y = 1\)
- \(y = 3\)
- \(x = 4\) (axis of rotation)

We will use the **method of cylindrical shells** since the axis of rotation is vertical and not one of the axes. The formula for the volume using cylindrical shells is:

\[
V = 2\pi \int_{y_1}^{y_2} \text{(radius)} \times \text{(height)} \, dy
\]

- The **radius** of a shell is the distance from the shell to the axis of rotation, which is \(4 - x\). In this case, \(x = \frac{2}{y}\), so the radius is \(4 - \frac{2}{y}\).
- The **height** of a shell is the function \(x = \frac{2}{y}\).

The limits of integration are from \(y = 1\) to \(y = 3\), as given by the bounds.

Thus, the volume is:

\[
V = 2\pi \int_1^3 \left(4 - \frac{2}{y}\right) \cdot \frac{2}{y} \, dy
\]

Now let's compute this integral.The volume of the solid generated by revolving the region around the line \(x = 4\) is:

\[
V = \frac{16\pi}{3} (\log(27) - 1)
\]

Would you like to go into more detail on any part of the solution or need further explanation?

Here are five related questions to explore:

1. How does the cylindrical shell method compare to the disk or washer method?
2. How would the solution change if the axis of rotation were \(x = 3\) instead of \(x = 4\)?
3. What are the key factors in choosing the appropriate method for solving volumes of revolution?
4. How would you approach this problem if the function was different, for example, \(xy = 4\)?
5. How does changing the bounds of \(y\) affect the final volume?

**Tip:** When using the method of cylindrical shells, the key is to set up the radius and height correctly based on the axis of rotation and the shape of the region.

Find the volume of the solid generated by revolving the region bounded by the equations about the line x = 4: xy = 2, y = 1, y = 3, and x = 4.

Learn how to calculate the volume of a solid generated by revolving the region bounded by the equations xy = 2, y = 1, y = 3, and x = 4 about the line x = 4. This problem involves using the cylindrical shell method and integration to find the volume.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation