The image you've uploaded depicts a graph where the shaded region is enclosed by the curve $y = 2 + \frac{x^2}{7}$ between $x = 0$ and $x = 1$, and the goal is to use the shell method to find the volume generated by revolving this shaded region about the y-axis.

To apply the shell method:

1. The formula for the shell method when revolving around the y-axis is:

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

Where:

• $f(x) = 2 + \frac{x^2}{7}$ is the function.
• $a = 0$ and $b = 1$ are the limits of integration.
• $x$ represents the radius of the cylindrical shell at a given point.

2. Substituting into the formula:

$V = 2 \pi \int_{0}^{1} x \left( 2 + \frac{x^2}{7} \right) \, dx$

Now we can simplify and compute this integral:

$V = 2 \pi \int_{0}^{1} \left( 2x + \frac{x^3}{7} \right) \, dx$

3. To compute the integral, integrate each term separately: $\int 2x \, dx = x^2$ $\int \frac{x^3}{7} \, dx = \frac{x^4}{28}$

Thus, the volume integral becomes:

$V = 2 \pi \left[ x^2 + \frac{x^4}{28} \right]_{0}^{1}$

4. Evaluating the definite integral:

$V = 2 \pi \left( 1^2 + \frac{1^4}{28} - (0^2 + \frac{0^4}{28}) \right)$ $V = 2 \pi \left( 1 + \frac{1}{28} \right)$ $V = 2 \pi \times \frac{29}{28}$ $V = \frac{58 \pi}{28} = \frac{29 \pi}{14}$

Thus, the volume generated by revolving the region about the y-axis is $\frac{29 \pi}{14}$.

Would you like any further details or clarifications on this problem?

Here are 5 follow-up questions that could help deepen your understanding:

1. How would the volume change if the region were revolved around the x-axis instead of the y-axis?
2. Can you explain the shell method in greater detail, specifically how the radius is derived for this problem?
3. How would you apply the disk method to solve the same problem?
4. What would happen if the limits of integration changed, such as using $x = 0$ to $x = 2$?
5. Could this problem be extended to find the volume of a more complex shape, such as one with multiple curves?

Tip: When using the shell method, always remember that the radius of each cylindrical shell is based on the horizontal distance from the axis of rotation (in this case, the y-axis).