Section 4.3 Fundamental Theorem of Calculus

Math 1231: Single-Variable Calculus

(Signed) Area function

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).

Signed area function: example

Fundamental Theorem of Calculus: Part I

The signed area function defined by f is an anti-derivative of f.

Examples

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

Examples

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).

Importance of FTC

Examples

More Examples

More Examples