Let \[R\] be the region enclosed by the \[y\]-axis, the line \[y=2\], the line \[y=5\], and the curve \[y=e^x\].

\[y\]\[x\]\[\blueD{y=e^x}\]\[\large R\]\[\small 2\]\[\small 5\]

A solid is generated by rotating \[R\] about the \[y\]-axis.

**Which one of the definite integrals gives the volume of the solid?**

To find the volume of the solid generated by rotating the region \( R \) around the \( y \)-axis, we use the method of cylindrical shells. Here's how we approach it:

### Step 1: Set Up the Shell Method Formula
The volume \( V \) of the solid can be calculated using the formula for the volume of a solid of revolution by cylindrical shells:

\[
V = 2\pi \int_{a}^{b} \text{(radius)} \times \text{(height)} \, dx
\]

### Step 2: Identify the Radius and Height
- **Radius:** The radius of a shell at a point \( x \) is simply \( x \).
- **Height:** The height of the shell is the difference between the upper and lower bounds of the region in the \( y \)-direction, which is \( e^x - 2 \) (since \( y = e^x \) and the line \( y = 2 \)).

### Step 3: Determine the Bounds
The bounds for \( x \) are determined by the intersection points of the line \( y = 2 \) and \( y = 5 \) with the curve \( y = e^x \). The curve intersects \( y = 2 \) at \( x = \ln(2) \) and \( y = 5 \) at \( x = \ln(5) \).

### Step 4: Write the Integral
Substituting the radius and height into the shell method formula:

\[
V = 2\pi \int_{\ln(2)}^{\ln(5)} x (e^x - 2) \, dx
\]

### Conclusion
The definite integral that gives the volume of the solid is:

\[
\boxed{2\pi \int_{\ln(2)}^{\ln(5)} x (e^x - 2) \, dx}
\]

Would you like any additional details or have any questions?

Here are some related questions:
1. How do you find the intersection points of a curve and a line?
2. What is the difference between the shell method and the disk/washer method?
3. How do you derive the formula for the volume of a solid of revolution?
4. What happens if the region is rotated about a different axis?
5. Can this method be applied if the region is not bounded by the y-axis?

**Tip:** The shell method is particularly useful when the region is easier to describe using vertical slices parallel to the axis of rotation.

Learn how to calculate the volume of a solid generated by rotating a region bounded by the y-axis, y=2, y=5, and the curve y=e^x around the y-axis. This problem involves using the cylindrical shells method in integral calculus. Suitable for college-level calculus students.

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