1. Definite Integration

2. As the number of rectangles increased, the approximation of the area under the curve approaches a value.

3. Definition
Note: The function f(x) must be continuous on the interval [a, b].

5. The numbers on the integral sign are called the limits of integration
is a definite integral
The numbers on the integral sign are called the limits of integration

6. Evaluating the Definite Integral
The definite integration results in a value.
e.g.1
Find the indefinite integral but omit C

7. Evaluating the Definite Integral
The definite integration results in a value.
e.g.1
Draw square brackets and hang the limits on the end

8. Evaluating the Definite Integral
The definite integration results in a value.
e.g.1
Replace x with
the top limit
the bottom limit

9. Evaluating the Definite Integral
The definite integration results in a value.
e.g.1
Subtract and evaluate

10. Evaluating the Definite Integral
The definite integration results in a value.
e.g.1
So,

11. SUMMARY
The method for evaluating the definite integral is:
Find the indefinite integral but omit C
Draw square brackets and hang the limits on the end
Replace x with
the top limit
the bottom limit
Subtract and evaluate

12. Evaluating the Definite Integral
e.g. 2 Find
Solution:
Indefinite integral but no C

13. Evaluating the Definite Integral
e.g. 2 Find
Solution:
Substitute for x:
top limit minus bottom limit
Simplify

14. Evaluating the Definite Integral
In this example, if we can’t use a calculator.
We must be very careful with the signs

15. Examples
1. Find
2. Find

16. Examples

17. Examples

18. Examples

19. Area & Integration

20. Area & Integration

21. Area & Integration

22. Area & Integration

23. Area & Integration

24. Area & Integration