1. Definition: Sec 5.2: THE DEFINITE INTEGRAL
the definite integral of f from a to b is
provided that this limit exists.
If it does exist, we say that is f integrable on [a,b]

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

3. Note 2: Sec 5.2: THE DEFINITE INTEGRAL
x is a dummy variable. We could use any variable

4. Note 3: Riemann sum Sec 5.2: THE DEFINITE INTEGRAL
Riemann sum is the sum of areas of rectangles.

5. Note 4: area under the curve Sec 5.2: THE DEFINITE INTEGRAL
Riemann sum is the sum of areas of rectangles.
area under the curve

6. Note 5: Sec 5.2: THE DEFINITE INTEGRAL
If takes on both positive and negative values,
the Riemann sum is the sum of the areas of the rectangles that lie above the -axis and the negatives of the areas of the rectangles that lie below the -axis (the areas of the gold rectangles minus the areas of the blue rectangles).
A definite integral can be interpreted as a net area, that is, a difference of areas:
where is the area of the region above the x-axis and below the graph of f , and is the area of the region below the x-axis and above the graph of f.

7. not all functions are integrable
Sec 5.2: THE DEFINITE INTEGRAL
Note 6:
not all functions are integrable
f(x) is cont [a,b]
f(x) has only finite number of removable discontinuities
f(x) has only finite number of jump discontinuities

8. Sec 5.2: THE DEFINITE INTEGRAL
f(x) is cont [a,b]
f(x) has only finite number of removable discontinuities
f(x) has only finite number of jump discontinuities

9. Sec 5.2: THE DEFINITE INTEGRAL
Note 7:
the limit in Definition 2 exists and gives the same value no matter how we choose the sample points

10. Sec 5.2: THE DEFINITE INTEGRAL

11. Sec 5.2: THE DEFINITE INTEGRAL
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12. Example: Sec 5.2: THE DEFINITE INTEGRAL Evaluate the Riemann sum for
taking the sample points to be right endpoints and a =0, b =3, and n = 6.
(b) Evaluate

13. 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.

14. Example: Example: Example: Sec 5.2: THE DEFINITE INTEGRAL
Set up an expression for
as a limit of sums
Evaluate the following integrals by interpreting each in terms of areas.
Example:
Evaluate the following integrals by interpreting each in terms of areas.

15. Sec 5.2: THE DEFINITE INTEGRAL

16. Sec 5.2: THE DEFINITE INTEGRAL

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. Note: Example: Sec 5.2: THE DEFINITE INTEGRAL
Property 1 says that the integral of a constant function is the constant times the length of the interval.
Example:
Use the properties of integrals to evaluate

21. Sec 5.2: THE DEFINITE INTEGRAL
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22. Sec 5.2: THE DEFINITE INTEGRAL

23. Sec 5.2: THE DEFINITE INTEGRAL
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24. Sec 5.2: THE DEFINITE INTEGRAL
Example:
Use Property 8 to estimate

25. SYMMETRY Sec 5.2: THE DEFINITE INTEGRAL
Suppose f is continuous on [-a, a] and even
Suppose f is continuous on [-a, a] and odd

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