Explain why or why not 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.

Verified step by step guidance

1. Recall the shell method is used to find volumes of solids of revolution by integrating cylindrical shells formed by revolving a region around an axis.
2. Understand that in the shell method, each shell is generated by revolving a vertical or horizontal strip around the axis of revolution.
3. Note that the axis of each cylindrical shell is actually perpendicular to the axis of revolution, because the shell is formed by rotating a line segment around that axis.
4. Therefore, the statement 'the axis of the cylindrical shells is parallel to the axis of revolution' is false.
5. A counterexample: If the axis of revolution is the x-axis, the shells are vertical cylinders whose axes are parallel to the y-axis, which is perpendicular to the x-axis.

Key Concepts

Shell Method for Volume Calculation

The shell method is a technique for finding the volume of a solid of revolution by integrating cylindrical shells. It involves slicing the solid into thin shells parallel to the axis of revolution and summing their volumes. Understanding how these shells are formed and their dimensions is crucial to applying the method correctly.

Axis of Revolution

The axis of revolution is the line about which a region is rotated to generate a solid. It serves as a reference for determining the shape and orientation of the solid and the shells or disks used in volume calculations. Recognizing the axis helps in setting up the integral and choosing the appropriate method.

Orientation of Cylindrical Shells Relative to the Axis

In the shell method, cylindrical shells are formed by revolving vertical or horizontal slices around the axis of revolution. Importantly, the axis of each shell is perpendicular, not parallel, to the axis of revolution. This orientation affects how the radius and height of each shell are defined in the integral.