Question

T/F: When finding the volume of a solid of revolution that was revolved around a vertical axis, the Shell Method integrates with respect to \(x\).

Short answer

False, the Shell Method integrates with respect to \(y\) for a vertical axis.

Step-by-step solution

Understanding the Shell Method

The Shell Method is used to find the volume of a solid of revolution. It involves integrating a function rotated around an axis. The idea is to calculate the volume by summing up cylindrical shells.

Identifying the Axis of Revolution

In this exercise, the solid is revolved around a vertical axis. The axis of revolution plays a critical role in determining the variable of integration in the Shell Method.

Determining the Variable of Integration

The Shell Method integrates with respect to the variable perpendicular to the axis of revolution. For a vertical axis, these are horizontal slices, which correspond to the variable \(y\). Hence, the integration will be with respect to \(y\).

Analysing the Given Statement

The given statement claims that the Shell Method integrates with respect to \(x\) for a vertical axis. From our analysis in Step 3, we know this is incorrect. The integration should be with respect to \(y\).

Key concepts

Volume of Solid of Revolution

The volume of a solid of revolution is a fascinating concept in calculus, involving the creation of a three-dimensional shape by rotating a two-dimensional area around an axis. Imagine taking a flat shape, like a rectangle or a semicircle, and spinning it around a line. This line is called the axis of revolution.

By doing this, you can create various solids like cylinders, cones, or tori, where the method chosen will help find the volume of these shapes.

• The most commonly used methods are the Disk and Washer Methods, and the Shell Method.
• Each method uses integration to calculate the volume of the solid formed after rotation.
• The choice of method often depends on the position of the axis of revolution and the shape of the function involved.

The Shell Method specifically helps when the traditional Disk or Washer Methods become cumbersome or complex. It's particularly useful when the rotation involves a vertical line (or axis). This method offers a more straightforward way to compute the volume of cylinders or shells built up by rotating the area.

Axis of Revolution

The axis of revolution is a crucial part of understanding volumes of solids formed by rotation. Think of it as the imaginary line you spin your shape around. It's like the pole on a playground around which children might spin.

The placement of this axis determines two main things:

• How you slice your solid: Depending on whether the axis is horizontal or vertical, your slices and thus your approach will differ.
• The method of integration chosen: This is especially true when choosing between the Disk, Washer, and Shell Methods.

In the Shell Method, if you revolve around a vertical axis, you integrate over horizontal slices, whereas a horizontal axis requires integration over vertical slices.

It's important to remember that whenever you choose or identify your axis of revolution, in most cases, it is positioned perpendicular to the slices you're considering in your volume calculation.

Cylindrical Shells

Cylindrical shells form the core idea of the Shell Method. Picture a series of nested tubes or coiled paper towel rolls stacked along the axis of revolution. Each cylindrical shell contributes a small amount to the overall volume.

The volume of a single shell can be calculated by determining:

• The circumference of the shell, which is dependent on the radius from the axis of revolution.
• The height of the function at that particular point, representing the thickness of each shell.
• The thickness of each shell, typically represented as an infinitesimal change in the variable, either \( \,dx\) or \( \,dy\).

When these elements are combined through integration, they sum up to give the total volume of the solid.

This method is especially efficient when the function being revolved is more easily described in terms of the radius and height along a given axis. For example, when a solid is rotated around a vertical axis, cylindrical shells stack horizontally, making the Shell Method practical and effective.

Integration with Respect to Variable

Integration is central to calculating volumes using the Shell Method, and the choice of variable significantly influences how easy or difficult the integration will be.

When considering a vertical axis of revolution, the process involves horizontal slices, leading to integration with respect to the variable \( \,y\), as the shells are stacked horizontally. Conversely, for a horizontal axis, integration is with respect to \( \,x\).

• This choice aligns the variable of integration perpendicular to the axis of revolution.
• Switching variables often necessitates solving the function for that variable, which involves rearranging equations as needed.

This principle ensures the integral precisely computes the sum of the volume elements, each represented by the incremental shell.

Correctly determining the variable is crucial in applying the Shell Method efficiently, particularly for complex-shaped areas or when the visual interpretation becomes challenging.