1. Sec 5.2: The Definite Integral
New Symbol
Definition:
the definite integral of ƒ over [a, b]
Example:
Find the definite integral of
ƒ(x) = x + 2 over [ -1, 1 ]
Solution:

2. The procedure of calculating an integral is called integration.
Sec 5.2: The Definite Integral
Note 1:
integrand
limits of integration
upper limit b
lower limit a
Integral sign
The dx simply indicates that the independent variable is x.
The procedure of calculating an integral is called integration.

3. the definite integral of f from a to b
Sec 5.2: The Definite Integral
Area under
the curve
Limit of the Riemann sum
If you are asked to find one of them choose the easiest one.
the definite integral of f from a to b
three sides of the same coin

4. Example: Sec 5.2: The Definite Integral
Evaluate the following integrals by interpreting each in terms of areas.

5. Sec 5.2: The Definite Integral
the definite integral can be interpreted as the area under the curve
definite integral has negative value
A definite integral can be interpreted as a net area, that is,a difference of areas:

6. Example: Sec 5.2: The Definite Integral
Evaluate the following integrals by interpreting each in terms of areas.

7. Example: Example: Sec 5.2: The Definite Integral
Evaluate the following integrals by interpreting each in terms of areas.
Example:
Evaluate the following integrals by interpreting each in terms of areas.

8. Sec 5.2: The Definite Integral
Term-121

9. the definite integral of f from a to b
Sec 5.2: The Definite Integral
Express the limit as a definite integral on the given interval.
the definite integral of f from a to b

10. Sec 5.2: The Definite Integral
Term-103

11. the definite integral of f from a to b
Sec 5.2: The Definite Integral
Area under
the curve
the definite integral of f from a to b
If you are asked to find one of them choose the easiest one.

12. Sec 5.2: The Definite Integral

13. Sec 5.2: The Definite Integral
Term-121

14. Sec 5.2: The Definite Integral
In fact, instead of using left endpoints or right endpoints, we could take the height of the ith rectangle to be the value of f at any number
in the ith subinterval
We call the numbers the sample points
Definition:
Definition:
Area =
Area =

15. Example: Sec 5.2: The Definite Integral Definition: Definition:
Find the Riemann sum for
ƒ(x) = x + 2 over [ 0, 5 ]
divided into

16. the definite integral of f from a to b
Sec 5.2: The Definite Integral
Area under
the curve
the definite integral of f from a to b
If you are asked to find one of them choose the easiest one.

17. Sec 5.2: The Definite Integral
Property (1)
Example:

18. Sec 5.2: The Definite Integral
Property (2)

19. Sec 5.2: The Definite Integral
Property (3)

20. Sec 5.2: The Definite Integral
Property (4)
Property (5)

21. Sec 5.2: The Definite Integral
Properties of the Integral

22. Sec 5.2: The Definite Integral

23. Sec 5.2: The Definite Integral
Definition:
Example:
provided that this limit exists
Find the definite integral of
ƒ(x) = x + 2 over [ -1, 1 ]
Solution:
Definition:
If the limit does exist, we say that the function f is integrable
the limit exist,
is integrable

24. Theorem: Sec 5.2: The Definite Integral
If f (x) is continuous on [a, b]
f (x) is integrable
Example:
is not integrable in [0, 1]
Remark
f(x) has only finite number of removable discontinuities
Remark
f(x) has only finite number of jump discontinuities

25. Sec 5.2: The Definite Integral

26. Sec 5.2: The Definite Integral
Term-091

27. Sec 5.2: The Definite Integral
Property

28. Sec 5.2: The Definite Integral
Property (6)

29. Sec 5.2: The Definite Integral
Property (7)

30. Sec 5.2: The Definite Integral

32. Sec 5.2: The Definite Integral

33. Sec 5.2: The Definite Integral

34. Sec 5.2: The Definite Integral
Term-082