2. 5.4 Fundamental Theorem of Calculus

3. Quick Review

4. Quick Review Solutions

5. What you’ll learn about Fundamental Theorem, Part 1 Graphing the Function Fundamental Theorem, Part 2 Area Connection Analyzing Antiderivatives Graphically Essential Questions How do we use the Fundamental Theorem of Calculus to solving area and volume problems?

6. The Fundamental Theorem of Calculus Every continuous function f is the derivative of some other function. Every continuous function has an antiderivative. The process of integration and differentiation are inverses of one another. Example Applying the Fundamental Theorem

7. Example The Fundamental Theorem with the Chain Rule

8. Example Variable Lower Limits of Integration 6. Construct a function of the form that satisfies the given conditions.

9. The Fundamental Theorem of Calculus, Part 2 If f is continuous at every point of [a, b], and if F is any antiderivative of f on [a, b], then Called the Integral Evaluation Theorem Any definite integral of any continuousfunction can be calculated without taking limits, without calculating Riemann sums, and often without effort - so long as an antiderivative f of can be found.f

10. Example Evaluating an Integral Evaluate each integral using Part 2 of the Fundamental Theorem.

11. How to Find Total Area Analytically How to Find Total Area Numerically

12. Example Find Total Area Find the total area of the region between the curve and the x-axis.

13. Pg. 302, 5.4 #1-47 odd