To find the volume of the solid generated by rotating the region $R$ around the line $x = -4$, we need to set up the integral using the method of cylindrical shells.

Step 1: Determine the formula for the volume using cylindrical shells

The formula for the volume using cylindrical shells when rotating around a vertical line (like $x = -4$) is given by:

$V = 2\pi \int_{y_{\text{min}}}^{y_{\text{max}}} \text{(radius)} \times \text{(height)} \, dy$

Step 2: Identify the radius and height

• Radius: The distance from the axis of rotation $x = -4$ to a point on the curve is given by the horizontal distance between $x = -4$ and $x$ on the curve. Thus, the radius $r(y) = |x + 4| = | \ln y + 4 | = \ln y + 4$.

• Height: The height of the shell is the difference between the $y$-values, which is $y$ (since we are using $y$ as the independent variable in this approach).

Step 3: Determine the limits of integration

The region $R$ is enclosed between $y = 1$ and $y = 4$, so the limits of integration are from $y = 1$ to $y = 4$.

Step 4: Set up the integral

Thus, the volume of the solid is:

$V = 2\pi \int_{1}^{4} (\ln y + 4) \cdot y \, dy$

However, the problem asks which of the definite integrals corresponds to the correct setup. The integral expression that matches the setup described above would be the following:

$V = 2\pi \int_{1}^{4} (\ln y + 4) \cdot y \, dy$

This represents the correct definite integral to calculate the volume of the solid generated by rotating the region $R$ about the line $x = -4$.

Would you like more details on this solution, or do you have any questions?

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Relative Questions:

1. How would the integral change if the rotation were around $y = 0$ instead?
2. What is the significance of the height function in the method of cylindrical shells?
3. How would you set up the integral if the region $R$ was rotated around the y-axis instead of $x = -4$?
4. Can the disk/washer method be applied to this problem? If yes, how?
5. What if the curve $y = e^x$ was replaced with $y = x^2$; how would the integral change?

Tip: Always visualize the region and the axis of rotation to help determine the correct method and setup for the integral.