1. The problem asks which method to use to find the volume of the solid formed by revolving the region bounded by $y=x^2$, $y=4x - x^2$, and $x=4$ about two different lines: $y=0$ (the x-axis) and $x=2$. 2. When revolving a region around a horizontal line like $y=0$, the **washer method** or **disk method** is typically used. This is because the cross-sections perpendicular to the axis of revolution are circular disks or washers. 3. When revolving around a vertical line like $x=2$, the **shell method** is often more convenient. This is because the shells are formed by vertical slices parallel to the axis of revolution. 4. Summary: - For revolution about $y=0$, use the **washer/disk method**. - For revolution about $x=2$, use the **shell method**. This choice is based on the orientation of the axis of revolution and the shape of the region.