4.4 Fundamental Theorem of Calculus Part 1

1. Photo by Vickie Kelly, 1998 Greg Kelly, Hanford High School, Richland, Washington 4.4 Fundamental Theorem of Calculus Part 1 Morro Rock, California

2. Photo by Vickie Kelly, 1998 Greg Kelly, Hanford High School, Richland, Washington 5.4 First Fundamental Theorem At this point, to compute this definite integral, we would need to: a) partition the interval into n subintervals b) write the summation for the area of each rectangle of width 2/n c) take the limit of the summation of the area of each rectangle as While we still need to know how to do this for the AP exam, there’s a MUCH easier way to do this! Morro Rock, California

3. The Fundamental Theorem of Calculus If f is continuous at every point of , and if F is any antiderivative of f on , then We can now easily compute more complicated definite integrals! We can thank Isaac Newton for this extremely powerful development in calculus!

4. Let’s look at a proof of this theorem: Let be the usual partition of [a, b]: If we write the difference F(b) – F(a) by subtracting one term from its previous term and adding like terms, we get: By the Mean Value Theorem (for derivatives), there must be a number ciin the interval such that:

5. Since F is the antiderivative of f, we know that: We can say that: So what happens to this summation as ? We get the definite integral: http://www.youtube.com/watch?v=usfiAsWR4qU Hooray, we know an easier way!!

6. Ex. 1 Evaluate: First find the antiderivative of: What about the C value?

7. Ex. 2 Evaluate: