Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.
y = ln 7x,  y = 2,  y = 4,  x = 0;   about the y-axis

To find the volume \( V \) of the solid obtained by rotating the region bounded by the curves about the y-axis, we will use the method of cylindrical shells.

The curves involved are:
- \( y = \ln(7x) \),
- \( y = 2 \),
- \( y = 4 \),
- \( x = 0 \).

### Step-by-Step Solution:

1. **Express \( x \) in terms of \( y \)**:  
   Start with the equation \( y = \ln(7x) \).  
   Solving for \( x \), we have:
   \[
   y = \ln(7x) \quad \Rightarrow \quad e^y = 7x \quad \Rightarrow \quad x = \frac{e^y}{7}.
   \]

2. **Set up the integral using cylindrical shells**:  
   The height of each shell is the difference between the upper and lower bounds of \( y \), and the radius is the distance from the y-axis (which is just \( x \)). The volume element \( dV \) for cylindrical shells is:
   \[
   dV = 2\pi \cdot (\text{radius}) \cdot (\text{height}) \cdot (\text{thickness}).
   \]
   Here:
   - Radius = \( x = \frac{e^y}{7} \),
   - Height = 1 (since we are rotating between \( y = 2 \) and \( y = 4 \)).

3. **Write the volume integral**:  
   The volume is obtained by integrating over \( y \) from 2 to 4:
   \[
   V = 2\pi \int_{2}^{4} \left( \frac{e^y}{7} \right) \, dy.
   \]

4. **Compute the integral**:  
   The integral of \( e^y \) is simply \( e^y \), so we have:
   \[
   V = 2\pi \cdot \frac{1}{7} \int_{2}^{4} e^y \, dy = \frac{2\pi}{7} \left[ e^y \right]_{2}^{4}.
   \]
   Now, evaluate the limits:
   \[
   V = \frac{2\pi}{7} \left( e^4 - e^2 \right).
   \]

Thus, the volume of the solid is:
\[
V = \frac{2\pi}{7} \left( e^4 - e^2 \right).
\]

Would you like me to evaluate this expression numerically or provide more details on any step?

### Related Questions:
1. How is the method of cylindrical shells different from the disk method?
2. Can we use the disk/washer method to solve this problem, and how would the approach change?
3. What are the key differences between rotating around the x-axis and the y-axis for volume calculations?
4. What would the volume be if the region were rotated around the line \( y = 1 \) instead of the y-axis?
5. How does the natural logarithmic function \( \ln(7x) \) impact the shape of the solid formed?

#### Tip:
Always ensure the bounds of integration are consistent with the variable of integration (here, in terms of \( y \)).

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = ln 7x, y = 2, y = 4, x = 0; about the y-axis

Learn how to calculate the volume of a solid formed by rotating the region bounded by y = ln(7x), y = 2, y = 4, and x = 0 about the y-axis. This detailed solution utilizes the cylindrical shell method and covers integration and logarithmic functions. 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