Common misconception on Volume of Revolution

These questions by students illustrate the common misconceptions on volume of revolution:

Question 1

Further queries by student

Misconception: Since the region is bounded on both sides of the Y-axis, the volume of revolution needs to be multiplied by 2 when integrate one side.

Clarification: The volume of revolution is the same whether one side of the region is rotated or both sides of the region are rotated. The swept volume of one side or both sides are the same.

Question 2

Solution

Notice that for part i, the volume needs to be multiplied by 2, whereas for part 2, there is no need.

Student’s queries on part ii:

Misconception: That the volume achieved by rotating only 180 degrees is half of the volume of rotating 360 degrees.

Clarification: Since the graph is bounded on both sides, just need to rotate by 180 degrees to obtain the full swept volume. If it is bounded by one side, then need to rotate by 360 degrees to obtain the full swept volume.