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.