1. Essential Question: How is a definite integral related to area ?

2. Definition of a Definite Integral A definite integral of a function y = f(x) on an interval [a, b] is the signed (  ) area between the curve and the x-axis.

3. Evaluating a Definite Integral (The Fundamental Theorem of Calculus, Part 1) If f is continuous on [a, b] and F(x) is any antiderivative of f(x), then You first take the antiderivative F(x) and then subtract the values at each endpoint (upper – lower)

4. ANTIDIFFERENTIATION

5. The following functions f(x), represent derivatives of some other functions F(x). Reverse the power rule process and find the functions F(x).

6. THE POWER RULE For any rational power r  -1, Notice, you add 1 first, then divide! (We will come back to the “c” later!)

7. ANTIDERIVATIVE OF A CONSTANT For any constant ‘k’,

8. Evaluate each definite integral

9. Ex 4: Find the area of the region between the graph of f(x) = x 2 – 4 and the x-axis from x = -2 to x = 2

10. Ex 5: Find the area of the region between the graph of f(x) = -x 2 + 4x – 3 and the x-axis from x = 0 to x = 2 The function crosses the x-axis within the given interval! (at x = 1)

11. Solution Ex 4: TOTAL AREA: 2 SQUARE UNITS

13. TRIG ANTIDERIVATIVES

14. LOG AND EXPONENTIAL

15. Constant Rule #1 Suppose that f(x) has an antiderivative. Then, for any constants a and b, You can ‘pull a constant out’ of an integral This is useful when your constant is something like 

16. Evaluate each definite integral

17. What’s up with the “c”?

18. Find the derivative of the following functions Why do we add “c”?

20. SIMPLE EXAMPLES: Evaluate each integral.

22. RANDOM EXAMPLES: Evaluate each integral.