To find the volume of the solid generated by revolving the region bounded by $y = 1 + \frac{x^2}{4}$, with bounds $x = 0$ and $x = 2$, about the y-axis, we can use the shell method.

The shell method formula for volume about the y-axis is:

$V = 2\pi \int_{a}^{b} x \cdot f(x) \, dx$ Where:

• $x$ is the radius of a shell.
• $f(x)$ is the height of the shell.
• $a$ and $b$ are the limits of integration (in this case, from $x = 0$ to $x = 2$).

Step 1: Identify the function and bounds

The given function is: $y = 1 + \frac{x^2}{4}$ We are rotating this about the y-axis, with bounds $x = 0$ to $x = 2$.

Thus, $f(x) = 1 + \frac{x^2}{4}$.

Step 2: Set up the integral using the shell method

We now set up the volume integral: $V = 2\pi \int_{0}^{2} x \cdot \left( 1 + \frac{x^2}{4} \right) \, dx$

Step 3: Solve the integral

Expand the integrand: $V = 2\pi \int_{0}^{2} \left( x + \frac{x^3}{4} \right) \, dx$ Now, integrate each term separately: $\int_{0}^{2} x \, dx = \frac{x^2}{2} \Bigg|_0^2 = \frac{2^2}{2} - \frac{0^2}{2} = 2$ $\int_{0}^{2} \frac{x^3}{4} \, dx = \frac{1}{4} \cdot \frac{x^4}{4} \Bigg|_0^2 = \frac{1}{16} \cdot (2^4 - 0^4) = \frac{1}{16} \cdot 16 = 1$

Step 4: Combine the results

Now, substitute the results of the integrals: $V = 2\pi \left( 2 + 1 \right) = 2\pi \times 3 = 6\pi$

Final Answer:

The volume of the solid is $\boxed{6\pi}$ cubic units.

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Would you like any further explanation on the shell method or related topics? Here are some related questions you might find interesting:

1. How does the shell method compare to the disk/washer method in finding volumes?
2. What if the region is revolved about the x-axis instead of the y-axis?
3. How do you apply the shell method to solids with different bounds or more complex functions?
4. What is the physical interpretation of using the shell method for volume?
5. Can the shell method be used in higher dimensions, and how?

Tip: When using the shell method, it's important to visualize the radius of each cylindrical shell and the height it spans in relation to the axis of rotation.