Definition: the definite integral of f from a to b is provided that this limit exists.

1. Sec 5.2: THE DEFINITE INTEGRAL Definition: the definite integral of f from a to b is provided that this limit exists. If it does exist, we say that is f integrableon [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. Sec 5.2: THE DEFINITE INTEGRAL Note 2: x is a dummy variable. We could use any variable

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

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

6. Sec 5.2: THE DEFINITE INTEGRAL Note 5: 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. 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

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

13. 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. Sec 5.2: THE DEFINITE INTEGRAL Example: Example: • 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.

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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 Note: 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

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

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

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