# Ch 12: Area Under the Curve and Integrals in AP Calculus: Help and Review

### About This Chapter

## Who's It For?

Anyone who needs help learning or mastering AP calculus material will benefit from taking this course. There is no faster or easier way to learn AP calculus. Among those who would benefit are:

- Students who have fallen behind in understanding area under the curve and integrals
- Students who struggle with learning disabilities or learning differences, including autism and ADHD
- Students who prefer multiple ways of learning math (visual or auditory)
- Students who have missed class time and need to catch up
- Students who need an efficient way to learn about area under the curve and integrals in AP calculus
- Students who struggle to understand their teachers
- Students who attend schools without extra math learning resources

## How It Works:

- Find videos in our course that cover what you need to learn or review.
- Press play and watch the video lesson.
- Refer to the video transcripts to reinforce your learning.
- Test your understanding of each lesson with short quizzes.
- Verify you're ready by completing the area under the curve and integrals for AP calculus chapter exam.

## Why it works:

**Study Efficiently:**Skip what you know; review what you don't.**Retain What You Learn:**Engaging animations and real-life examples make topics easy to grasp.**Be Ready on Test Day:**Use the area under the curve and integrals in AP calculus chapter exam to be prepared.**Get Extra Support:**Ask our subject-matter experts any question about area under the curve and integrals. They're here to help!**Study With Flexibility:**Watch videos on any web-ready device.

## Students will review:

This chapter helps students review the concepts in an area under the curve and integrals unit of a standard AP Calculus AB and BC course. Topics covered include:

- Using Riemann sums for functions and graphs
- Writing out the trapezoid rule
- Finding the limits of Reimann sums
- Using Riemann sums to calculate integrals
- Using the average value theorem
- Stating the relationship between indefinite integrals and derivatives
- Using the fundamental theorem of calculus
- Finding the arc length of a function

### 1. How to Use Riemann Sums for Functions and Graphs

Find out how Riemann sums can be used to calculate multiple areas efficiently. In this lesson, you'll learn how this can come in handy for irregular areas and how you can put it to use.

### 2. How to Identify and Draw Left, Right and Middle Riemann Sums

To overestimate or underestimate? In this lesson, you will draw Riemann rectangles so the right side, left side or middle of the rectangle hits the curve.

### 3. What is the Trapezoid Rule?

In this lesson, you'll move beyond simple rectangles to estimate the area under a curve. Get more sophisticated with your approximations and use trapezoids instead of those pesky rectangles.

### 4. How to Find the Limits of Riemann Sums

What would happen if you could draw an infinite number of infinitesimally thin rectangles? You'd get the exact area under a curve! Define the Holy Grail of calculus, the integral, in this lesson.

### 5. Definite Integrals: Definition

Explore how driving backwards takes you where you've already been as we define definite integrals. This lesson will also teach you the relationship between definite integrals and Riemann sums. Then, discover how an integral changes when it is above and below the x-axis.

### 6. How to Use Riemann Sums to Calculate Integrals

As a new property owner, you might relish mowing your lawn. Up and down your property you mow and measure out small sections to find the area of your property. In this lesson, you will discover what a Riemann sum approach is and how to calculate an estimated area using multiple slices.

### 7. Linear Properties of Definite Integrals

If you're having integration problems, this lesson will relate integrals to everyday driving examples. We'll review a few linear properties of definite integrals while practicing with some problems.

### 8. Average Value Theorem

If you know you've gone 120 miles in 2 hours, you're averaging 60 mph. But what if you know your velocity at every point in time and not how far you've gone? In this lesson, learn how to calculate average values using integrals.

### 9. The Fundamental Theorem of Calculus

The fundamental theorem of calculus is one of the most important equations in math. In this lesson we start to explore what the ubiquitous FTOC means as we careen down the road at 30 mph.

### 10. Indefinite Integrals as Anti Derivatives

What does an anti-derivative have to do with a derivative? Is a definite integral a self-confident version of an indefinite integral? Learn how to define these in this lesson.

### 11. 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.

### 12. Riemann Sums: Formula & Concept

When we want to find the area of the irregular space between the x-axis and a continuous graph, we can't just draw a rectangle in there and take the area; it doesn't fit. If we draw lots of little rectangles, though, it comes closer. If we keep using more and more rectangles that are smaller and smaller, we'll keep getting closer and closer to the true area.

### 13. U Substitution: Examples & Concept

In this lesson, you will learn how to use the substitution technique for integration and also learn to recognize the types of problems with which you can use this method.

### 14. Double Integrals: Applications & Examples

Double integrals extend the possibilities of one-dimensional integration. In this lesson, we will focus on the application of the double integral for finding enclosed area, volume under a surface, mass specified with a surface density, first and second moments, and the center of mass.

### 15. Double Integration: Method, Formulas & Examples

In this lesson, we explore the method of double integration, which is useful in finding certain areas, volumes, and masses of objects. Note: knowledge of (single) integrals is a must before tackling double integrals!

### 16. Improper Integral: Definition & Examples

When integrating a function, infinity might appear in the limits of integration. The function itself could also go to infinity. These types of integrals are called improper integrals, which we will explore in this lesson.

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### Other Chapters

Other chapters within the AP Calculus AB & BC: Help and Review course

- Functions in AP Calculus: Help and Review
- Graphing and Functions in AP Calculus: Help and Review
- Sequences and Series in AP Calculus: Help and Review
- Limits in AP Calculus: Help and Review
- Continuity in AP Calculus: Help and Review
- Exponentials and Logarithms in AP Calculus: Help and Review
- Exponents and Polynomials in AP Calculus: Help and Review
- Applications of Derivatives in AP Calculus: Help and Review
- Calculating Derivatives & Derivative Rules in AP Calculus: Help & Review
- Calculus - Derivatives Calculations & Rules: Help & Review
- Differential Equations in AP Calculus: Help and Review
- L'Hopital's Rule & Graphing Derivatives: Help & Review
- Integration Applications in AP Calculus: Help and Review
- Rate of Change in AP Calculus: Help and Review
- Geometry and Trigonometry in AP Calculus: Help and Review
- How to Use Scientific Calculators for AP Calculus: Help and Review