About This Chapter
Integral Calculus - Chapter Summary
Our instructors will define integrals and introduce you to Riemann sums and how to use them in your calculations. You will learn to solve and evaluate integrals using substitution and the fundamental theorem of calculus. The concepts of anti-derivatives and integration by parts are also introduced.
Your study continues with the information needed to analyze curves and find the arc length of a function. Once you have reviewed this chapter, you should be able to:
- Understand the relationship between definite integrals and Riemann sums
- Calculate estimated area using multiple slices
- Simplify complicated integrals
- Apply the fundamental theorem of calculus
- Break down big integrals using integration by parts
- Solve a variety of integration problems
- Find the length of a curve
- Differentiate between monotonicity and concavity
Our mobile-friendly lessons and quizzes have been created by professional instructors to make sure your learning experience is simple and quick. Use the quizzes and chapter tests to see how well you understand the subject matter. If you'd like to watch just a section of one of the video lessons, use the video tabs feature in the Timeline to jump directly to that information. If you have questions as you work through the chapter, contact one of our instructors for more help.
1. 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.
2. 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.
3. How to Solve Integrals Using Substitution
Some integrals are as easy as riding a bike. But more often, integrals can look like deformed bikes from Mars in the year 3000. In this lesson, you will learn how to transform these scary-looking integrals into simpler ones that really are as easy as riding a bike.
4. Evaluating Definite Integrals Using the Fundamental Theorem
The fundamental theorem of calculus makes finding your definite integral almost a piece of cake. See how the definite integral becomes a subtraction problem after applying the fundamental theorem of calculus.
5. Anti-Derivatives: Calculating Indefinite Integrals of Polynomials
If you throw a ball in the air, the path that it takes is a polynomial. In this lesson, learn how to integrate these fantastic functions by putting together your knowledge of the fundamental theorem of calculus and your ability to differentiate polynomial functions.
6. Using Integration By Parts
Your mother may have warned you not to bite off more than you can chew. The same thing is true with integration. In this lesson, learn how integration by parts can help you split a big interval into bite-sized pieces!
7. 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.
8. Integration Problems in Calculus: Solutions & Examples
In this lesson, you'll learn about the different types of integration problems you may encounter. You'll see how to solve each type and learn about the rules of integration that will help you.
9. 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.
10. Using Monotonicity & Concavity to Analyze Curves
There are all kinds of graphs, and we have various ways of describing our curves. We have words such as monotonic, concave up, and concave down. Watch this video lesson to learn how to identify these kinds of graphs.
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