Question

Determine whether the following statements are true and give an explanation or counterexample.

A) When using the shell​ method, the axis of the cylindrical shells is parallel to the axis of revolution.
B) If a region is revolved about the​ y-axis, then the shell method must be used.
C) If a region is revolved about the​ x-axis, then in principle it is possible to use the​ disk/washer method and integrate with respect to x or the shell method and integrate with respect to y.

Answer 1

The statements are explained and counterexamples are provided.

Step-by-step explanation:

A) When using the shell method, the axis of the cylindrical shells is parallel to the axis of revolution.

B) If a region is revolved about the​ y-axis, then the shell method must be used.

C) If a region is revolved about the​ x-axis, then in principle it is possible to use the​ disk/washer method and integrate with respect to x or the shell method and integrate with respect to y.

Answer 2

A) When using the shell​ method, the axis of the cylindrical shells is parallel to the axis of revolution. True.

The Shell method is a technique used to find the volume of a solid of revolution. Here, we take thin shells with axis coinciding with the axis about which the region whose volume is to be found, is revolved.

B) If a region is revolved about the​ y-axis, then the shell method must be used. False.

This method can be used with any axis of rotation.

C) If a region is revolved about the​ x-axis, then in principle it is possible to use the​ disk/washer method and integrate with respect to x or the shell method and integrate with respect to y. True.

The washer method uses thin disks with infinite width but the shell method uses thin concentric shells with infinite width about the axis of revolution. So, the statement is true.