1. Section 4.3 Fundamental Theorem of Calculus Math 1231: Single-Variable Calculus

2. (Signed) Area function

3. Signed area function: example Example If f is a function whose graph is shown in the Figure and, find the values of g(0), g(1), g(2), g(3), g(4), g(5).

4. Signed area function: example

5. Fundamental Theorem of Calculus: Part I The signed area function defined by f is an anti-derivative of f.

6. Examples

7. Fundamental Theorem of Calculus: Part II The Fundamental Theorem of Calculus, Part 2 If f is continuous on [a, b], then where F is any anti-derivative of f. Property

8. Examples

9. Differentiation and Integration: Inverse processes The Fundamental Theorem of Calculus If f is continuous on [a, b], then 1. : If f is integrated and then is differentiated, we arrive back to f. 2. : If we take a function F, first differentiate it then integrate it, we arrive back to F, but in the form of F(b) – F(a).

10. Importance of FTC

11. Examples

12. More Examples