When to Use Cylindrical Shells?

You should use the method of cylindrical shells primarily when calculating the radii required for the washer or disk method becomes overly complicated.

Why Cylindrical Shells?

The method of cylindrical shells is a powerful technique for finding the volume of a solid of revolution. It is particularly advantageous in situations where the traditional disk or washer method proves difficult.

The Challenge with Washers/Disks

The washer or disk method calculates volume by summing up infinitesimally thin disks or washers perpendicular to the axis of revolution. This means you integrate with respect to the variable along the axis of revolution (e.g., dy if revolving around the y-axis).

Sometimes, the function defining the boundary of the region is given in a form (like y = f(x)) that makes it hard or impossible to solve for the other variable (x in terms of y).

According to the reference: "Method of Cylindrical Shells is used when it becomes complicated to compute inner and outer radii of a washer." This complication often arises when you would need to solve for x in terms of y (or vice-versa) to set up the washer integral, as mentioned in the example for Figure 1 in the reference.

How Cylindrical Shells Offer a Solution

Cylindrical shells calculate volume by summing up the surface area of thin cylindrical shells parallel to the axis of revolution. This allows you to integrate with respect to the variable perpendicular to the axis of revolution (e.g., dx if revolving around the y-axis).

This is often much simpler when your function is given as y = f(x) and you are revolving around a vertical axis (like the y-axis or a line x = k), or when your function is x = g(y) and you are revolving around a horizontal axis (like the x-axis or a line y = k).

Common Scenarios Favoring Cylindrical Shells

Here are some specific situations where cylindrical shells are frequently the easier approach:

• Revolving a region defined by y = f(x) around a vertical axis (y-axis or x = k): The washer method would require solving for x in terms of y. The shell method uses dx and the function f(x) directly for the height of the shell.
• Revolving a region defined by x = g(y) around a horizontal axis (x-axis or y = k): The washer method would require solving for y in terms of x. The shell method uses dy and the function g(y) directly for the radius of the shell.
• When solving for the inverse function is impossible or difficult: Some functions simply cannot be easily inverted (e.g., y = x³ + x). If revolving such a function necessitates inversion for the washer method, cylindrical shells are the preferred choice.
• When the region boundary changes: For some complex regions, using the washer method might require splitting the integral into multiple parts due to changes in which function forms the inner or outer radius. Cylindrical shells can sometimes handle such regions with a single integral.

In summary: If setting up the disk or washer method requires substantial algebraic effort to redefine your function or splits the problem into multiple integrals, consider using cylindrical shells. They often provide a more straightforward path by allowing integration with respect to the variable that corresponds directly to the form of your given function.