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The image presents a problem involving the calculation of volumes generated by rotating regions \( R_1 \) and \( R_2 \) around various axes. The specific questions ask for the volume of the regions generated by rotation around different lines.

### Regions:
- \( R_1 \) is bounded by \( x = 4y \), \( y = 0 \), and \( x = 8 \).
- \( R_2 \) is bounded by \( x = 4y \), \( y = \frac{1}{2}x \), and \( y = 2 \).

### Rotation Lines:
1. **Question 7:** Volume of region \( R_1 \) rotated around the x-axis (OA).
2. **Question 8:** Volume of region \( R_1 \) rotated around the line BC.
3. **Question 9:** Volume of region \( R_2 \) rotated around the x-axis (OA).
4. **Question 10:** Volume of region \( R_2 \) rotated around the y-axis (OC).

Let’s now find the volumes generated by the rotation of each region.

### 1. Volume of \( R_1 \) around \( OA \) (x-axis):
For the region \( R_1 \), the function representing the boundary is \( x = 4y \), rewritten as \( y = \frac{x}{4} \). We will use the **disk method** for rotation around the x-axis. 

The volume formula is:
\[
V = \pi \int_{a}^{b} \left[f(x)\right]^2 dx
\]
where \( f(x) = \frac{x}{4} \), with limits of integration from \( x = 0 \) to \( x = 8 \).

### 2. Volume of \( R_1 \) around \( BC \) (line \( x = 8 \)):
This involves the **shell method**, since the axis of rotation is parallel to the y-axis. The volume formula for shells is:
\[
V = 2\pi \int_{a}^{b} (radius)(height) \, dy
\]
Here, the radius is the distance from \( x = 8 \) and the height is the function of \( y \), which is \( x = 4y \).

### 3. Volume of \( R_2 \) around \( OA \) (x-axis):
For region \( R_2 \), the upper boundary is \( y = \frac{1}{2}x \), and the lower boundary is \( y = \frac{x}{4} \). We can use the **washer method** for rotation around the x-axis. The volume formula is:
\[
V = \pi \int_{a}^{b} \left[\left(f(x)\right)^2 - \left(g(x)\right)^2\right] \, dx
\]
where \( f(x) = \frac{1}{2}x \) and \( g(x) = \frac{x}{4} \), and limits from \( x = 0 \) to \( x = 8 \).

### 4. Volume of \( R_2 \) around \( OC \) (y-axis):
For this case, we use the **shell method** for rotation around the y-axis. The volume formula is:
\[
V = 2\pi \int_{a}^{b} (radius)(height) \, dx
\]
where the radius is \( x \), and the height is the difference between the two functions, \( \frac{1}{2}x \) and \( \frac{x}{4} \).

Let me calculate the volumes.Here are the calculated volumes for the first two cases:

- **Question 7**: The volume of region \( R_1 \) rotated around the x-axis (OA) is:
  \[
  V = \frac{32\pi}{3} \, \text{units}^3
  \]

- **Question 9**: The volume of region \( R_2 \) rotated around the x-axis (OA) is:
  \[
  V = 32\pi \, \text{units}^3
  \]

Next, I'll calculate the volumes for rotations around BC and OC.Here are the calculated volumes for the remaining cases:

- **Question 8**: The volume of region \( R_1 \) rotated around the line BC is:
  \[
  V = \frac{32\pi}{3} \, \text{units}^3
  \]

- **Question 10**: The volume of region \( R_2 \) rotated around the y-axis (OC) is:
  \[
  V = \frac{256\pi}{3} \, \text{units}^3
  \]

Would you like further details on any of these calculations?

Here are some related questions to expand your understanding:
1. How does the disk method differ from the shell method for volume calculation?
2. Why is the washer method used when rotating region \( R_2 \) around the x-axis?
3. What are the implications of rotating around different lines, such as OA vs. BC?
4. How would the results change if the region were rotated around a line offset from both axes?
5. What are the limits of integration, and how are they chosen for different rotation methods?

**Tip**: For volumes of rotation, always carefully check the axis of rotation to decide between the disk, washer, or shell methods.

Veja a figura e encontre o volume gerado pela rotação da região indicada em torno da reta dada.

Solve a problem involving the calculation of volumes generated by rotating regions around various axes using methods like the disk, washer, and shell. Key concepts include geometry, integral calculus, and volume of solids of revolution.

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