# Ch 14: Integration Applications in AP Calculus: Tutoring Solution

### About This Chapter

## How it works:

- Begin your assignment or other AP calculus work.
- Identify the integration applications concepts that you're stuck on.
- Find fun videos on the topics you need to understand.
- Press play, watch and learn!
- Complete the quizzes to test your understanding.
- As needed, submit a question to one of our instructors for personalized support.

## Who's it for?

This chapter of our AP calculus tutoring solution will benefit any student who is trying to learn about integration applications for AP calculus and earn better grades. This resource can help students including those who:

- Struggle with understanding how to use integration in motion, find simple areas and area between functions with integration, calculate volumes using integrals or any other integration applications in AP calculus topic
- Have limited time for studying
- Want a cost effective way to supplement their math learning
- Prefer learning math visually
- Find themselves failing or close to failing their integration applications in AP calculus unit
- Cope with ADD or ADHD
- Want to get ahead in AP calculus
- Don't have access to their math teacher outside of class

## Why it works:

**Engaging Tutors:**We make learning about integration applications for AP calculus simple and fun.**Cost Efficient:**For less than 20% of the cost of a private tutor, you'll have unlimited access 24/7.**Consistent High Quality:**Unlike a live calculus tutor, these video lessons are thoroughly reviewed.**Convenient:**Imagine a tutor as portable as your laptop, tablet or smartphone. Learn about integration applications on the go!**Learn at Your Pace:**You can pause and rewatch lessons as often as you'd like, until you master the material.

## Learning Objectives

- Determine location as a function of time using integration and dynamic motion.
- Use root finding and integration to calculate simple areas.
- Calculate the area between two functions using integration.
- Use single integrals to calculate volumes.
- Determine volumes of revolution using integration.

### 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. Solving the Integral of cos(2x)

In this lesson, we will find the integral of cos(2x) using integration by substitution in a step-by-step process. We will then go on to check our work using derivatives and the chain rule for derivatives.

### 7. Solving the Integral of cos(x)

In this lesson, we will see how to find the integral of cos(x) using well known derivatives and the fundamental theorem of calculus. We will also examine some other trigonometric functions and how we can use this theorem to see a connection between their derivatives and integrals.

### 8. Solving the Integral of ln(x)

Finding the integral of ln(x) requires the process of integration by parts. This lesson will show how to find the integral of ln(x) using integration by parts and explain how and why integration by parts works.

### 9. Finding the Antiderivative of 1/cos(x)

It's handy to know the antiderivative of certain functions. In this lesson, we will calculate the antiderivative of 1 / cos(x). First, we will use a trigonometric identity to see how to use a well known formula to find this antiderivative and then we will take a look at where this formula comes from.

### 10. Finding the Integral of e^x

In this lesson, we show how to find the integral of e^x by considering the series expansion for the exponential function and the fundamental theorem of calculus. A numerical application is included which highlights the integration result.

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### Other Chapters

Other chapters within the AP Calculus AB & BC: Tutoring Solution course

- Functions in AP Calculus: Tutoring Solution
- Graphing and Functions in AP Calculus: Tutoring Solution
- Sequences and Series in AP Calculus: Tutoring Solution
- Limits in AP Calculus: Tutoring Solution
- Continuity in AP Calculus: Tutoring Solution
- Exponentials and Logarithms in AP Calculus: Tutoring Solution
- Exponents and Polynomials in AP Calculus: Tutoring Solution
- Applications of Derivatives in AP Calculus: Tutoring Solution
- AP Calculus - Derivatives Calculations & Rules: Tutoring Solution
- Differential Equations in AP Calculus: Tutoring Solution
- Area Under the Curve and Integrals in AP Calculus: Tutoring Solution
- AP Calculus - Graphing Derivatives & L'Hopital's Rule: Tutoring Solution
- AP Calculus - Integration: Tutoring Solution
- Rate of Change in AP Calculus: Tutoring Solution
- Geometry and Trigonometry in AP Calculus: Tutoring Solution
- How to Use Scientific Calculators for AP Calculus: Tutoring Solution