To solve this problem using the method of cylindrical shells, let's break it down step by step.

Step 1: Understanding the Problem

We are given the region bounded by the following curves:

• $x = 4 + y^2$
• $x = 0$ (which is the y-axis)
• $y = 1$
• $y = 2$

We are asked to find the volume of the solid obtained by rotating this region about the $x$-axis.

Step 2: Set Up the Cylindrical Shells

The method of cylindrical shells involves integrating along the $y$-axis, since we are rotating around the $x$-axis.

For a given $y$, the shell formed by rotating the region around the $x$-axis has:

• Radius ($r$): The distance from the $x$-axis to the point on the shell, which is just $y$.

  $r = y$

• Height ($h$): The height of the shell is the horizontal distance between the curves $x = 4 + y^2$ and $x = 0$ (the y-axis).

  $h = (4 + y^2) - 0 = 4 + y^2$

So, the radius $r = y$ and the height $h = 4 + y^2$.

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Would you like to proceed with the next steps or have any specific questions about this step?

Here are five relative questions to expand your understanding:

1. How do you determine the limits of integration for this problem?
2. What is the formula for the volume using the method of cylindrical shells?
3. How would the setup change if the region were rotated around the $y$-axis instead?
4. How does the orientation of the strip (horizontal vs. vertical) affect the cylindrical shells method?
5. What would the height and radius be if the curves were different?

Tip: When using the method of cylindrical shells, always check whether it's more convenient to integrate along the $x$-axis or the $y$-axis. This choice can simplify your calculations significantly.