To find the volume of the solid generated by revolving the region bounded by the curve $y = x$ from $x = 0$ to $x = 2$ around the x-axis using the shell method, follow these steps:

1. Set up the shell method formula

The formula for the volume using the shell method when revolving around the x-axis is:

$V = 2\pi \int_{a}^{b} (radius)(height) \, dx$

Here:

• The radius is the distance from the x-axis to the strip, which is simply $y$.
• The height of each cylindrical shell is the function of $y$, which is $x = y$.

2. Define the limits of integration

Since we're integrating with respect to $y$, we need to change the limits to match the $y$-coordinates:

• At $x = 0$, $y = 0$.
• At $x = 2$, $y = 2$.

Thus, the limits of integration are from $y = 0$ to $y = 2$.

3. Write the integral for the volume

The height is given by $x = y$, and the radius is $y$. Thus, the integral becomes:

$V = 2\pi \int_{0}^{2} y(y) \, dy$ $V = 2\pi \int_{0}^{2} y^2 \, dy$

4. Evaluate the integral

Now, we evaluate the definite integral:

$\int y^2 \, dy = \frac{y^3}{3}$

Apply the limits of integration:

$V = 2\pi \left[ \frac{y^3}{3} \right]_{0}^{2}$

$V = 2\pi \left( \frac{2^3}{3} - \frac{0^3}{3} \right)$

$V = 2\pi \left( \frac{8}{3} - 0 \right)$

$V = 2\pi \cdot \frac{8}{3}$

$V = \frac{16\pi}{3}$

Final Answer:

The volume of the solid generated by revolving the region bounded by $y = x$ from $x = 0$ to $x = 2$ about the x-axis is:

$V = \frac{16\pi}{3}$

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Do you want more details or have any further questions on this problem?

Here are 5 related questions you might consider:

1. How would the shell method differ if revolving around the y-axis?
2. What is the difference between the shell method and the disk method?
3. Can the same integral be solved using the disk/washer method for this problem?
4. How do you interpret the result geometrically in terms of volume?
5. What would change if the region was $y = x^2$ instead of $y = x$?

Tip: Always check the axis of rotation carefully, as it determines the setup for the shell method or disk method.