can you make the multiplier pi instead of 2 pi

Certainly! If we adjust the problem so that the multiplier is \( \pi \) instead of \( 2\pi \), it suggests we're using a different method—specifically, the **disk/washer method**—to find the volume of the solid.

### Disk/Washer Method

In this method, the volume \( V \) of the solid of revolution is given by:

\[
V = \pi \int_{c}^{d} \left[\text{outer radius}^2 - \text{inner radius}^2\right] \, dy
\]

Here, the outer radius and inner radius are functions of \( y \), and the bounds \( c \) and \( d \) are the \( y \)-values between which we are integrating.

### Step 1: Identify the Radii
- **Outer Radius:** Since we are revolving around the \( y \)-axis, the outer radius corresponds to the \( x \)-value on the curve \( y = e^x \). Solving for \( x \), we get \( x = \ln(y) \).
- **Inner Radius:** The inner radius corresponds to the \( x \)-value where \( y = 2 \), which is a vertical line at \( x = 0 \) (i.e., the \( y \)-axis).

### Step 2: Determine the Bounds
The region \( R \) is bounded between \( y = 2 \) and \( y = 5 \).

### Step 3: Set Up the Integral
Substituting the radii and bounds into the formula:

\[
V = \pi \int_{2}^{5} \left[\ln(y)^2 - 0^2\right] \, dy
\]

Simplifying, we get:

\[
V = \pi \int_{2}^{5} \ln(y)^2 \, dy
\]

### Conclusion
The definite integral that gives the volume of the solid with a \( \pi \) multiplier is:

\[
\boxed{\pi \int_{2}^{5} \ln(y)^2 \, dy}
\]

Would you like any additional details or have any questions?

Here are some related questions:
1. How do you compute the integral of \( \ln(y)^2 \)?
2. What are the steps to derive the disk/washer method?
3. How does the choice of method (shell vs. disk/washer) affect the complexity of the problem?
4. Can you apply the disk/washer method if the region is rotated about a different vertical line?
5. How do you determine which method (shell or disk/washer) is more appropriate for a given problem?

**Tip:** The disk/washer method is especially useful when the region is easier to describe using horizontal slices perpendicular to the axis of rotation.

Learn how to calculate the volume of a solid of revolution using the disk/washer method with a π multiplier instead of 2π. Detailed step-by-step solution includes identifying radii, setting up the integral, and applying bounds. Suitable for college-level calculus students.

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