Problem 2 - Calculus Early Transcendentals

Question

Fill in the blanks: A region \(R\) is revolved about the \(y\) -axis. The volume of the resulting solid could (in principle) be found by using the disk/washer method and integrating with respect to ___________ to using the shell method and integrating with respect to ___________.

Step-by-Step Solution

Answer

A region R is revolved about the \(y\)-axis. The volume of the resulting solid could (in principle) be found by using the disk/washer method and integrating with respect to \(y\) or using the shell method and integrating with respect to \(x\).

Step 1: 1. Disk/Washer Method

When using the disk/washer method to find the volume of a solid formed by revolving a region R around the y-axis, the slices we take are perpendicular to the y-axis. In this situation, we integrate with respect to y.

Step 2: 2. Shell Method

In the shell method, the cylinders we form are parallel to the axis of revolution, which in this case is the y-axis. So, we will be integrating with respect to x. Answer: A region R is revolved about the \(y\)-axis. The volume of the resulting solid could (in principle) be found by using the disk/washer method and integrating with respect to \(\boxed{y}\) to using the shell method and integrating with respect to \(\boxed{x}\).

Key Concepts

Disk/Washer Method

The disk/washer method is a popular technique for finding the volume of a solid of revolution. Imagine taking thin disk-shaped slices of the solid perpendicular to the axis about which the region is revolved. If we're revolving around the \(y\)-axis, the disks are also perpendicular to the \(y\)-axis.

This method works by adding up the volumes of these thin disks. Each disk has a tiny height, \(\Delta y\), and a radius that may vary depending on the function or region you are rotating.

The formula for the volume of one of these disks is \(\pi (\text{radius})^2 \cdot \Delta y\). To find the total volume, we integrate this expression with respect to \(y\):

• Calculate the radius of each disk as a function of \(y\).
• Set up the integral from the lower to the upper bounds of \(y\).
• Evaluate the integral to find the volume.

Using this method can be particularly useful when the region is bounded by functions that are easily expressed as \(y\) values.

Shell Method

The shell method offers an alternative way to find the volume of a solid of revolution. This method uses cylindrical shells instead of disks. When the region is revolved around the \(y\)-axis, each shell is parallel to the axis and coaxial with it.

Picture a stack of cylindrical shells that, together, make up the solid. To quantify the volume, think of slicing the region into vertical strips and then rotating those strips around the \(y\)-axis.

For each shell, the radius is the \(x\)-value of the strip, and the height is given by the function defining the region:

• The formula for the volume of a shell is \(2\pi (\text{radius}) \times (\text{height}) \times \Delta x\).
• Integrate this expression with respect to \(x\).
• The integral spans from the leftmost to the rightmost points of \(x\).

The shell method is especially helpful when dealing with functions that describe \(x\) in terms of \(y\), or when simplifying the setup.

Integration with Respect to Axis

Understanding the axis of integration is critical in determining which method to use for calculating the volume of a solid of revolution. Depending on the method chosen, you will integrate with respect to different variables.

When using the disk/washer method:

• Integrate with respect to \(y\) if revolving around the \(y\)-axis.
• This involves slicing the region horizontally.

For the shell method:

• Integrate with respect to \(x\) when revolving around the \(y\)-axis.
• This involves considering the vertical strips.

Being clear about the axis of rotation and the direction of slices helps in setting up the integral correctly. It results in more straightforward calculations and ensures that the geometry of the solid of revolution is taken into account properly.