Find volume of solid bounded by given curves about the specified lines: y=ln(x), y=1, y=4, x=0; about y axis

To find the volume of the solid formed by revolving the region bounded by the curves \( y = \ln(x) \), \( y = 1 \), \( y = 4 \), and \( x = 0 \) about the y-axis, we will use the **method of cylindrical shells**.

### Step 1: Analyze the region

The region is bounded by:
- The curve \( y = \ln(x) \)
- The vertical lines \( y = 1 \) and \( y = 4 \)
- The vertical line \( x = 0 \) (the y-axis)

### Step 2: Set up the formula for the volume

Using the method of cylindrical shells, the formula for the volume of a solid of revolution about the y-axis is:

\[
V = 2\pi \int_{y_1}^{y_2} x(y) \cdot h(y) \, dy
\]

Where:
- \( x(y) \) is the expression for \( x \) in terms of \( y \).
- \( h(y) \) is the height of the shell, which is the difference between the two bounding functions for \( x \).

From the equation \( y = \ln(x) \), we can express \( x \) as a function of \( y \):

\[
x = e^y
\]

The limits of integration are from \( y = 1 \) to \( y = 4 \), because those are the bounds in the \( y \)-direction.

### Step 3: Set up the integral

Now, we can plug this into the formula:

\[
V = 2\pi \int_{1}^{4} e^y \cdot 1 \, dy
\]

### Step 4: Evaluate the integral

\[
V = 2\pi \int_{1}^{4} e^y \, dy
\]

The integral of \( e^y \) is simply \( e^y \), so we get:

\[
V = 2\pi \left[ e^y \right]_{1}^{4}
\]

\[
V = 2\pi \left( e^4 - e^1 \right)
\]

\[
V = 2\pi \left( e^4 - e \right)
\]

### Final Answer:

The volume of the solid is:

\[
V = 2\pi \left( e^4 - e \right)
\]

You can compute this numerically if needed, but this is the exact form of the volume.

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Do you want me to expand on any of the steps, or would you like further clarifications?

Here are 5 related questions for practice:
1. How do you apply the method of washers to this problem?
2. What is the volume if the region is revolved about the line \( x = 1 \)?
3. How do you handle cases where the region is revolved around a horizontal line?
4. What is the effect of changing the upper bound to \( y = 5 \)?
5. How does the volume change if \( x = 1 \) is used as a boundary?

**Tip:** Always ensure your functions are expressed in terms of the correct variable for the chosen axis of revolution.

Find the volume of the solid bounded by the curves: y = ln(x), y = 1, y = 4, x = 0; about the y-axis.

Learn how to calculate the volume of the solid formed by rotating the region bounded by y = ln(x), y = 1, y = 4, and x = 0 about the y-axis. This solution uses the method of cylindrical shells and provides a step-by-step guide. Suitable for college-level calculus students.

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