# Ch 16: Applications of Integrals

### About This Chapter

## Applications of Integrals - Chapter Summary and Learning Objectives

We all memorized equations such as A = 1/2 base *x* height, A = ? *x* r^2, and V=base *x* height *x* length in basic arithmetic classes, but those and other area equations, simple and complex, are derived from working with integrals in calculus. Explore area and volume in more complicated shapes in this chapter on using integrals in our everyday life. The video lessons are brief and full of expert instruction you're unlikely to forget. Use the quizzes and chapter test to gauge understanding and figure out which topics you may want to revisit for further practice. In this chapter you will discover more about:

- Dynamic motion and integration
- Calculating simple areas with roots and integration
- Finding the area between functions
- Determining volumes with single integrals
- Calculating volumes of revolution through integration
- Determining a function's arc length

Video | Objective |
---|---|

Integration and Dynamic Motion | This video describes how to determine position or velocity at a given point using integration of a function f(x) wherein position is a function of time. |

How to find Simple Areas With Root Finding and Integration | Learn to use the root of an equation to determine the area of a space created within the intersections of several functions in this lesson. |

How to Find Area Between Functions with Integration | Instructors describe how to use integration to find areas with complex boundaries. |

How to Calculate Volumes Using Single Integrals | This video explores how the concept behind Riemann sums can be used to 'slice' an object and find the sum of the slices. |

How to Find Volumes of Revolution With Integration | Do some more slicing as you learn how to find volumes of objects which are symmetrical along an axis in this video. |

How to Find the Arc Length of a Function | In this video our instructors explain how to calculate the distance between two points when the path is an arc rather than a straight line. |

### 1. Integration and Dynamic Motion

This lesson uses driving to demonstrate how graphing can help you use calculus to figure out how fast you were going at a given time. Using this lesson, you can learn how to integrate your velocity to find your position.

### 2. How to Find Simple Areas With Root Finding and Integration

Combine your calculus tools in this lesson! Find the area enclosed by multiple arbitrary functions by first finding the root of an equation and then integrating over the resulting range.

### 3. How to Find Area Between Functions With Integration

Sometimes you aren't looking for the area under the curve; after all, not every region is between a curve and axis! In this lesson, learn how to find areas between curves as well as areas with complicated boundaries.

### 4. How to Calculate Volumes Using Single Integrals

Ever wonder where the equation for the volume of a cone comes from? Or the equation for the volume of a sphere? In this lesson, learn how to use a slicing technique to find the volume of a region by solving a single integral.

### 5. How to Find Volumes of Revolution With Integration

Some shapes look the same as you rotate them, like the body of a football. In this lesson, learn how to find the volumes of shapes that have symmetry around an axis using the volume of revolution integration technique.

### 6. How to Find the Arc Length of a Function

You don't always walk in a straight line. Sometimes, you want to know the distance between two points when the path is curved. In this lesson, you'll learn about finding the length of a curve.

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### Other Chapters

Other chapters within the AP Calculus BC: Exam Prep course

- Graph Basics
- The Basics of Functions
- How to Graph Functions
- Limits of Functions
- Function Continuity
- Understanding Exponentials & Logarithms
- Using Exponents and Polynomials
- Parametric, Polar and Vector Functions
- Properties of Derivatives
- The Derivative at a Point
- The Derivative as a Function
- Second Derivatives
- Derivative Applications
- Finding Derivatives
- Properties of Definite Integrals
- Using the Fundamental Theorem of Calculus
- Applying Integration Techniques
- Approximation of Definite Integrals
- Understanding Sequences & Series
- Series of Constants
- Taylor Series
- Using a Scientific Calculator for Calculus
- AP Calculus BC Flashcards