About This Chapter
Below is a sample breakdown of the Area Under the Curve and Integrals chapter into a 5-day school week. Based on the pace of your course, you may need to adapt the lesson plan to fit your needs.
|Day||Topics||Key Terms and Concepts Covered|
|Monday|| Summation Notation and Mathematical Series; |
How to Use Riemann Sums for Functions and Graphs;
How to Identify and Draw Left, Right and Middle Riemann Sums
| The use of sigma notation; |
The process of using sum notation;
The process of constructing Riemann rectangles
|Tuesday|| What is the Trapezoid Rule?; |
How to Find the Limits of Riemann Sums;
Definite Integrals: Definition
| Estimating area with trapezoids; |
An outline of this process;
Examples of definite integrals
|Wednesday|| How to Use Riemann Sums to Calculate Integrals; |
Linear Properties of Definite Integrals
| The process of calculating area; |
An explanation of the zero integral property and the backward property
|Thursday|| Average Value Theorem; |
The Fundamental Theorem of Calculus
| The process of finding an average value; |
An explanation of this theorem
|Friday|| Indefinite Integrals as Anti Derivatives; |
How to Find the Arc Length of a Function
| Using the fundamental theorem; |
Finding distance with the Pythagorean theorem
1. Summation Notation and Mathematical Series
The capital Greek letter sigma might be most recognizable as a common symbol in a fraternity or sorority, but it's used for some pretty cool math tricks too. In this lesson, learn how to use sigma for things like determining how many seats are in Michigan Stadium.
2. 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.
3. 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.
4. 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.
5. 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.
6. 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.
7. 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.
8. 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.
9. 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.
10. 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.
11. 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.
12. 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 Calculus Syllabus Resource & Lesson Plans course
- Graphing & Functions: Calculus Lesson Plans
- Continuity: Calculus Lesson Plans
- Geometry & Trigonometry: Calculus Lesson Plans
- Using Scientific Calculators: Calculus Lesson Plans
- Limits: Calculus Lesson Plans
- Rate of Change: Calculus Lesson Plans
- Calculating Derivatives: Calculus Lesson Plans
- Derivative Graphs & L'Hopital's Rule: Calculus Lesson Plans
- Applications of Derivatives: Calculus Lesson Plans
- Integration: Calculus Lesson Plans
- Integration Applications: Calculus Lesson Plans
- Differential Equations: Calculus Lesson Plans