1. 5.c – The Fundamental Theorem of Calculus and Definite Integrals

2. Examples The definite integral of f (x) from x = a to x = b is denoted f(x) is called the integrand, a the lower limit of integration, and b the upper limit of integration.

3. The First Fundamental Theorem of Calculus

4. 4 Basic Properties of the Indefinite Integral Let a, b, and c be constants and f and g be continuous functions on [a, b].

5. Examples Evaluate:

6. 6 Definite Integrals With The Substitution Rule If u = g(x) is a differentiable function whose range is an interval I and f is continuous on I, then Properties of Odd and Even: Suppose f is continuous on [– a, a].

7. 7 Examples - Evaluate Evaluate by changing your limits of integration to values that are in terms of u.

8. First Fundamental Theorem of Calculus (Alternate Definition) We’ve shown that represents the general antiderivative of f with respect to x. It follows that the derivative of the antiderivative should return the original function (that is, the integrand). t is called a dummy variable. Upper Limit Must Be Variable Part Lower Limit Must Be Numerical Part

9. Examples Evaluate the following derivatives.

10. Examples Let u is some function of x. Use WolframAlpha to determine the following:

11. First Fundamental Theorem of (Generalized) If f is continuous on [a, b] and u is an unknown, differentiable function of x, then

12. Examples Evaluate