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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 integrable on [a,b] Sec 5.2: THE DEFINITE INTEGRAL

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

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

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

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

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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.

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Note 6: Sec 5.2: THE DEFINITE INTEGRAL 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

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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

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

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Sec 5.2: THE DEFINITE INTEGRAL

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Term-092

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

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

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Sec 5.2: THE DEFINITE INTEGRAL

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Midpoint Rule Sec 5.2: THE DEFINITE INTEGRAL We often choose the sample point to be the right endpoint of the i-th subinterval because it is convenient for computing the limit. But if the purpose is to find an approximation to an integral, it is usually better to choose to be the midpoint of the interval, which we denote by.

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Sec 5.2: THE DEFINITE INTEGRAL

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

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Sec 5.2: THE DEFINITE INTEGRAL Property (2)

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Sec 5.2: THE DEFINITE INTEGRAL Property (3)

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Example: 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. Use the properties of integrals to evaluate

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Sec 5.2: THE DEFINITE INTEGRAL Term-091

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Sec 5.2: THE DEFINITE INTEGRAL

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Term-092

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

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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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Sec 5.2: THE DEFINITE INTEGRAL Term-102

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Sec 5.2: THE DEFINITE INTEGRAL Term-102

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Sec 5.2: THE DEFINITE INTEGRAL Term-091

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Sec 5.2: THE DEFINITE INTEGRAL Term-091

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Sec 5.2: THE DEFINITE INTEGRAL

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Term-102

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Sec 5.2: THE DEFINITE INTEGRAL Term-082

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Sec 5.2: THE DEFINITE INTEGRAL Term-082

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Sec 5.2: THE DEFINITE INTEGRAL Term-103

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Sec 5.2: THE DEFINITE INTEGRAL Term-103

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Sec 5.2: THE DEFINITE INTEGRAL Term-103

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Sec 5.2: THE DEFINITE INTEGRAL Term-103

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Sec 5.2: THE DEFINITE INTEGRAL Term-092

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Sec 5.2: THE DEFINITE INTEGRAL Term-082

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