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]

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.

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

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

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

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.

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

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

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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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

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.

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.

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Sec 5.2: THE DEFINITE INTEGRAL Property (1) Example:

Sec 5.2: THE DEFINITE INTEGRAL Property (2)

Sec 5.2: THE DEFINITE INTEGRAL Property (3)

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

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Sec 5.2: THE DEFINITE INTEGRAL

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

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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