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Section 4.3

Section 4.3. Definition of a Riemann Sum:. Definition of a Definite Integral. The definite integral of f from a to b is defined as a is called the lower limit of integration and b the upper limit of integration. Theorem: Continuity Implies Integrability.

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Section 4.3

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  1. Section 4.3 Definition of a Riemann Sum:

  2. Definition of a Definite Integral The definite integral of f from a to b is defined as a is called the lower limit of integration and b the upper limit of integration.

  3. Theorem: Continuity Implies Integrability If a function fis continuous on the closed interval [a, b], then f is integrable on [a, b].

  4. The Definite Integral as Area Under Curve If f is continuous and nonnegative on the closed interval [a, b], then the area under the curve and above the x-axis from x = a to x = b is given by

  5. Two Special Definite Integrals • If f is defined at x = a, then • If f is integrable on [a, b], then

  6. Additive Integral Property If f is integrable on the three closed intervals determined by a, b, and c, then

  7. Properties of Definite Integrals If f and g are integrable on [a, b] and k is a constant, then the functions of kf and f ± g are integrable on [a, b], and

  8. One Last Thing Three important relationships are as follows: 1. 2. 3.

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