key term - $V = ext{pi} ext{int}_{a}^{b} ([f(x)]^2 - [g(x)]^2) ext{dx}$

Definition

This term represents the formula for calculating the volume of a solid generated by rotating a region bounded by the functions $f(x)$ and $g(x)$ about the $x$-axis over the interval $[a, b]$. It is a fundamental concept in the topic of determining volumes by slicing.

5 Must Know Facts For Your Next Test

1. The term $V = ext{pi} ext{int}_{a}^{b} ([f(x)]^2 - [g(x)]^2) ext{dx}$ is used to calculate the volume of a solid generated by rotating a region bounded by the functions $f(x)$ and $g(x)$ about the $x$-axis over the interval $[a, b]$.
2. The integral $ ext{int}_{a}^{b} ([f(x)]^2 - [g(x)]^2) ext{dx}$ represents the cross-sectional area of the solid at each point along the $x$-axis, and the factor $ ext{pi}$ converts this area into the volume of the solid of revolution.
3. The Disk Method, Washer Method, and Cylindrical Shells are all techniques that utilize the formula $V = ext{pi} ext{int}_{a}^{b} ([f(x)]^2 - [g(x)]^2) ext{dx}$ to find the volume of a solid of revolution.
4. The choice of which method to use depends on the specific functions $f(x)$ and $g(x)$ that define the region being rotated, as well as the ease of integration for each approach.
5. Understanding the derivation and application of the formula $V = ext{pi} ext{int}_{a}^{b} ([f(x)]^2 - [g(x)]^2) ext{dx}$ is crucial for solving volume problems involving solids of revolution.

Review Questions

• Explain the relationship between the formula $V = ext{pi} ext{int}_{a}^{b} ([f(x)]^2 - [g(x)]^2) ext{dx}$ and the Disk Method for determining the volume of a solid of revolution.
  • The formula $V = ext{pi} ext{int}_{a}^{b} ([f(x)]^2 - [g(x)]^2) ext{dx}$ is directly related to the Disk Method for finding the volume of a solid of revolution. In the Disk Method, the solid is divided into thin circular disks, and the volume of each disk is calculated using the formula for the area of a circle, $ ext{pi} r^2$, where $r$ is the radius of the disk. The formula $V = ext{pi} ext{int}_{a}^{b} ([f(x)]^2 - [g(x)]^2) ext{dx}$ captures this idea by integrating the difference between the squares of the bounding functions $f(x)$ and $g(x)$, which represent the radii of the disks at each point along the $x$-axis.
• Describe how the formula $V = ext{pi} ext{int}_{a}^{b} ([f(x)]^2 - [g(x)]^2) ext{dx}$ can be used to determine the volume of a solid generated by rotating a region bounded by $f(x)$ and $g(x)$ about the $y$-axis.
  • To use the formula $V = ext{pi} ext{int}_{a}^{b} ([f(x)]^2 - [g(x)]^2) ext{dx}$ to find the volume of a solid generated by rotating a region bounded by $f(x)$ and $g(x)$ about the $y$-axis, the roles of $f(x)$ and $g(x)$ must be reversed. In this case, the formula becomes $V = ext{pi} ext{int}_{c}^{d} ([h(y)]^2 - [k(y)]^2) ext{dy}$, where $h(y)$ and $k(y)$ are the bounding functions in the $y$-direction, and the limits of integration are $[c, d]$. This adjustment is necessary because the cross-sectional area of the solid at each point along the $y$-axis is now given by the difference between the squares of the bounding functions, rather than along the $x$-axis.
• Analyze the key differences between using the formula $V = ext{pi} ext{int}_{a}^{b} ([f(x)]^2 - [g(x)]^2) ext{dx}$ and the Cylindrical Shells method for finding the volume of a solid of revolution.
  • The primary difference between using the formula $V = ext{pi} ext{int}_{a}^{b} ([f(x)]^2 - [g(x)]^2) ext{dx}$ and the Cylindrical Shells method for finding the volume of a solid of revolution lies in the approach. The formula $V = ext{pi} ext{int}_{a}^{b} ([f(x)]^2 - [g(x)]^2) ext{dx}$ focuses on calculating the cross-sectional area of the solid at each point along the $x$-axis and then integrating this area to find the total volume. In contrast, the Cylindrical Shells method divides the solid into thin cylindrical shells and calculates the volume of each shell individually before summing them. The choice between these two methods depends on the specific functions $f(x)$ and $g(x)$ that define the region being rotated, as well as the ease of integration for each approach.

Related terms

Disk Method:

A technique for finding the volume of a solid of revolution by dividing the solid into thin circular disks and summing their volumes.

Washer Method:

A method for finding the volume of a solid of revolution by dividing the solid into thin circular washers and summing their volumes.

Cylindrical Shells: A method for finding the volume of a solid of revolution by dividing the solid into thin cylindrical shells and summing their volumes.