About This Chapter
Who's it for?
This unit of our AP Calculus Homeschool course will benefit any student who is trying to learn about area under the curve and integrals. There is no faster or easier way to learn about calculus. Among those who would benefit are:
- Students who require an efficient, self-paced course of study to learn to calculate an integral with Riemann sums.
- Homeschool parents looking to spend less time preparing lessons and more time teaching.
- Homeschool parents who need a calculus curriculum that appeals to multiple learning types (visual or auditory).
- Gifted students and students with learning differences.
How it works:
- Students watch a short, fun video lesson that covers a specific unit topic.
- Students and parents can refer to the video transcripts to reinforce learning.
- Short quizzes and an Area Under the Curve and Integrals unit exam confirm understanding or identify any topics that require review.
Area Under the Curve and Integrals Unit Objectives:
- Find the limits of Riemann sums.
- Define the Trapezoid Rule.
- Learn to identify and draw types of sums.
- Understand linear properties of definite integrals.
- Learn to use the average value theorem.
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.
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Other chapters within the AP Calculus AB & BC: Homeschool Curriculum course
- Functions - AP Calculus: Homeschool Curriculum
- Graphing & Functions - AP Calculus: Homeschool Curriculum
- Sequences & Series - AP Calculus: Homeschool Curriculum
- Limits - AP Calculus: Homeschool Curriculum
- Continuity - AP Calculus: Homeschool Curriculum
- Exponentials & Logarithms: Homeschool Curriculum
- Exponents & Polynomials - AP Calculus: Homeschool Curriculum
- Applications of Derivatives - AP Calculus: Homeschool Curriculum
- Calculating Derivatives & Derivative Rules - AP Calculus: Homeschool Curriculum
- Differential Equations - AP Calculus: Homeschool Curriculum
- Graphing Derivatives & L'Hopital's Rule - AP Calculus: Homeschool Curriculum
- Integration & Integration Techniques - AP Calculus: Homeschool Curriculum
- Integration Applications - AP Calculus: Homeschool Curriculum
- Rate of Change - AP Calculus: Homeschool Curriculum
- Geometry and Trigonometry - AP Calculus: Homeschool Curriculum
- Using Scientific Calculators - AP Calculus: Homeschool Curriculum