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The Fundamental Theorem of Calculus Inverse Operations

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Fundamental Theorem of Calculus Discovered independently by Gottfried Liebnitz and Isaac Newton Informally states that differentiation and definite integration are inverse operations.

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Fundamental 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

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Guidelines for Using the Fundamental Theorem of Calculus 1. Provided you can find an antiderivative of f, you now have a way to evaluate a definite integral without having to use the limit of a sum. 2. When applying the Fundamental Theorem of Calculus, the following notation is used

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Guidelines It is not necessary to include a constant of integration C in the antiderivative because they cancel out when you subtract.

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Evaluating a Definite Integral

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Evaluate the Definite Integral

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Definite Integral Involving Absolute Value Evaluate

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Definite Integral Involving Absolute Value

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Using the Fundamental Theorem to Find Area Find the area of the region bounded by the graph of y = 2x 3 – 3x + 2, the x- axis, and the vertical lines x = 0 and x = 2

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Using the Fundamental Theorem to Find Area

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The Mean Value Theorem for 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

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Average Value of a Function This is just another way to write the Mean Value Theorem (mean = average in mathematics) If f is integrable on the closed interval [a,b], then the average value of f on the interval is

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Average Value of a Function

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Finding the Average Value of a Function Find the average value of f(x) = sin x on the interval [0, ]

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Force The force F (in newtons) of a hydraulic cylinder in a press is proportional to the square of sec x, where x is the distance (in meters) that the cylinder is extended in its cycle. The domain of F is [0, /3] and F(0) = 500.

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Force (a) Find F as a function of x. F(x) = 500 sec 2 x (b) Find the average force exerted by the press over the interval [0, /3]

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Force

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Second Fundamental Theorem of Calculus If f is continuous on an open interval I containing a, then, for every x in the interval,

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Using the Second Fundamental Theorem of Calculus Evaluate

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Second Fundamental Theorem of Calculus Find F’(x) of

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