1. Rotate about the x-axis:
Radius: y
Height: The function describing the shaded region in terms of y.
2. Rotate about the y-axis:
Radius: x
Height: The function describing the shaded region in terms of x.
3. Rotate about the line x = 3:
Radius: ∣x−3∣ (Distance from x to the axis of rotation)
Height: The function describing the shaded region in terms of x.
4. Rotate about the line y = 2:
Radius: ∣y−2∣ (Distance from y to the axis of rotation)
Height: The function describing the shaded region in terms of y.
5. Rotating a rectangle will always form a cylinder.
Radius: The shorter side of the rectangle
Height: The longer side of the rectangle
When revolving a shaded region around a specific axis to form a solid of revolution, the resulting geometry and dimensions depend on the axis of rotation.
In the first scenario, rotating about the x-axis, the solid formed is a disk. The radius of this disk is given by the y-coordinate of the function describing the shaded region, and the height is determined by the function itself.
For rotation about the y-axis, a washer-shaped solid is generated. The radius of this washer is defined by the x-coordinate of the function representing the shaded region, while the height is dictated by the function.
When rotating about the line x = 3, the resulting solid takes the form of a cylindrical shell. The radius is determined by the distance of the x-coordinate from the axis of rotation (in this case, 3), and the height is specified by the function representing the shaded region.
Similarly, rotating about the line y = 2 also results in a cylindrical shell. The radius is now determined by the distance of the y-coordinate from the axis of rotation (2), and the height is defined by the function describing the shaded region.
Finally, when a rectangle is rotated, it consistently forms a cylinder. The radius of this cylinder is the shorter side of the rectangle, while the height is the longer side of the rectangle. This holds true regardless of the specific axis of rotation.