for Teachers for Schools for Enterprise
for Teachers for Schools for Enterprise
Course Navigator
Calculating Integrals of Simple ShapesNext Lesson 

Evaluating Definite Integrals Using the Fundamental Theorem

Chapter 16 /  Lesson 2   Transcript
Try it risk-free for 30 days

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Calculating Integrals of Simple Shapes

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
 Replay
Your next lesson will play in 10 seconds
  • 0:00 A Definite Integral
  • 0:34 The Fundamental…
  • 1:17 Example 1
  • 2:30 Example 2
  • 3:16 Lesson Summary
Add to Add to Add to

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Login or Sign up

Timeline
Autoplay
Autoplay
Speed

Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

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.

A Definite Integral

Let's get right into this lesson about definite integrals. A definite integral is the integral of a function from a starting point to an end point. You'll see little numbers at the top and bottom of your integral sign telling you where your integration begins and where it ends. The bottom value gives you your starting point and the top value gives you your end point.

This is an example of a general, definite integral:

definite integral

When you read it, you read it saying: 'the definite integral of the function f of x from point a to point b with respect to x.

The Fundamental Theorem of Calculus

To help us evaluate our definite integrals we have a fundamental theorem of calculus. This theorem tells us that the definite integral of f or x from point a to point b, with respect to x, is equal to the integral of f of x evaluated at point b minus the integral of f of x evaluated at point a.

definite integral

Mathematically we write this where our big F stands for the integral. This fundamental theorem of calculus actually helps us a lot when it comes to evaluating our definite integral. It actually makes it very easy for us to do, especially when we've memorized our common integrals as well as our common integration rules, such as the power rule.

Let's take a look at a couple of examples to see how easy it is.

Example 1

Evaluate:

definite integral

To evaluate this definite integral, we first find the integral of 3x^2, it is 3x^3 / 3 = x^3 + C. We have the constant of integration C when we take the general integral. When we take the definite integral we end up subtracting C from C, which essentially removes it, so we don't worry about it when we work with definite integrals.

Now we can apply the fundamental theorem of calculus. We first evaluate the x^3 + C at the point x = 3. Plugging in 3 for x we get: 3^3 + C = 27 +C.

Now we evaluate the x^3 + C at the point x = 1. Plugging in 1 for x we get: 1^3 + C = 1 +C.

Now subtracting the (1 + C) from the (27 + C) we get: (27 + C) - (1 + C) = 27 - 1 = 26.

To unlock this lesson you must be a Study.com Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use Study.com

Become a Study.com member and start learning now.
Become a Member  Back
What teachers are saying about Study.com
Try it risk-free for 30 days
Description

Earning College Credit

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Research Schools, Degrees & Careers

Get the unbiased info you need to find the right school.

Browse Articles By Category

Browse an area of study or degree level.

Recommended Articles
You are viewing lesson Lesson 2 in chapter 16 of the course:

 Back To Course

AP Calculus AB: Exam Prep

21 chapters | 138 lessons | 6 flashcard sets

Ch 16. Using the Fundamental Theorem of Calculus
Ch 17. Understanding & Applying...

Evaluating Definite Integrals Using the Fundamental Theorem Related Study Materials

Create an account to start this course today
Try it risk-free for 30 days!
Create An Account

Explore our library of over 75,000 lessons

Browse by subject
Support