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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
![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](http://images.slideplayer.com./23/6879993/slides/slide_3.jpg "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
![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](http://images.slideplayer.com./23/6879993/slides/slide_13.jpg "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
![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](http://images.slideplayer.com./23/6879993/slides/slide_14.jpg "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, ]
![Finding the Average Value of a Function Find the average value of f(x) = sin x on the interval [0, ]](http://images.slideplayer.com./23/6879993/slides/slide_16.jpg "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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