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The Fundamental Theorem of Calculus

# The Fundamental Theorem of Calculus - The Fundamental...

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The Fundamental The Fundamental Theorem of Calculus Theorem of Calculus Section 4.4

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Thm. 4.9 The Fundamental Thm. 4.9 The Fundamental Theorem of Calculus Theorem of Calculus If a function f is continuous on the closed interval [a, b] and F is an antiderivative of f on the interval [a, b], then a b f(x) dx = F(b) – F(a)
The following notation is convenient to use when evaluating a definite integral: It is not necessary to include a constant of integration C because it will always “cancel out” in a definite integral. a b f(x) dx = F(x) a b

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Thm. 4.10 Mean Value Thm. for Thm. 4.10 Mean Value Thm. for Integrals Integrals If f is continuous on the closed interval [a, b], then there exists a number c in the closed interval [a, b] such that a b f(x) dx = f(c) (b – a)
Mean Value Rectangle: f(c) a c b (b-a)

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Defn. of the Average Value of a Defn. of the Average Value of a Function on an Interval Function on an Interval If f is integrable on the closed interval [a, b], then the average value of f on the interval is 1 b - a a b f(x) dx Look at Ex. 4 pg. 278
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