region bounded by the curve
2 1
y x  
and the line
3
y x
 
is revolved about the x-axis to
generate the solid. Find the volume of the solid.

To find the volume of the solid generated by revolving the region bounded by the curve \( y = x + 2 \) and the line \( y = 3 - x \) about the x-axis, we'll use the method of disks.

### Step 1: Find the Points of Intersection
First, we need to determine the points of intersection between the curve \( y = x + 2 \) and the line \( y = 3 - x \).

Set the equations equal to each other:
\[
x + 2 = 3 - x
\]

Solve for \( x \):
\[
2x = 1 \quad \Rightarrow \quad x = \frac{1}{2}
\]

Now, substitute \( x = \frac{1}{2} \) back into either equation to find the corresponding \( y \)-value:
\[
y = \frac{1}{2} + 2 = \frac{5}{2}
\]

Thus, the points of intersection are \( \left(\frac{1}{2}, \frac{5}{2}\right) \).

### Step 2: Determine

Calculate the volume of a solid generated by revolving the region bounded by the curve y = x + 2 and the line y = 3 - x around the x-axis. Detailed step-by-step solution using the disks method.

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