Use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the x-axis.
y = x from 0 to 2

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:

\[
V = \frac{16\pi}{3}
\]

---

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.

Use the shell method to write and evaluate the definite integral that represents the volume of the solid generated by revolving the plane region about the x-axis. y = x from 0 to 2

Learn how to find the volume of a solid generated by revolving the region bounded by y = x from x = 0 to x = 2 around the x-axis using the shell method. This solution involves calculus concepts such as the definite integral and volume of solids of revolution, suitable for college-level calculus students.

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