About This Chapter
Area Under the Curve and Integrals
If you had to write out every term for all of the sums that we deal with in calculus, you'd have time for little else. Luckily, we can use sum notation to write these sums more compactly and save time. We'll teach you how to translate between sum notation and expanded equations, especially when it comes to Riemann sums. These sums are more than just numbers; we can use them to find the areas under all kinds of functions and graphs, even irregular areas that you may see in everyday life.
Taking these sums of areas is all about rectangles and choosing endpoints. If that sounds strange, don't worry; we'll show you how to identify and draw different types of sums, including left, middle and right sums. There's bound to be to be some overestimation and underestimation of areas when we use mere rectangles, but you'll improve upon this method with the trapezoid rule. Our lessons will strive to give you an intuitive understanding of calculus concepts as you see how shapes can fit a curve.
Riemann sums are only the beginning. We want to lead you to the holy grail of calculus: the integral. You'll learn how to wield these powerful tools when you find out how to take the limits of Riemann sums. Definite integrals will be our starting point; you'll see how a Riemann sum approach can be used to calculate them and come away with an understanding of their linear properties.
We can think about functions in parts and we can think about them globally. As you progress through this calculus section, you'll see how the mean value theorem allows you to do both and get a sense of its possible applications. We'll also define the fundamental theorem of calculus, so you won't want to miss that lesson. Finally, we'll examine anti-derivatives to explore the vital relationship among indefinite integrals, definite integrals and derivatives.
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 Math 104: Calculus course
- Graphing and Functions
- Vectors in Calculus
- Geometry and Trigonometry
- How to Use a Scientific Calculator
- Rate of Change
- Calculating Derivatives and Derivative Rules
- Graphing Derivatives and L'Hopital's Rule
- Applications of Derivatives
- Integration and Integration Techniques
- Integration Applications
- Differential Equations
- Studying for Math 104