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4.4 The Fundamental Theorem of Calculus If a function is continuous on the closed interval [a, b], then where F is any function that F’(x) = f(x) x in [a, b].

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Examples 1 – 0 = 1

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Integration with Absolute Value We need to rewrite the integral into two parts.

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Ex. Find the area of the region bounded by y = 2x 2 – 3x + 2, the x-axis, x = 0, and x = 2.

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The Mean Value Theorem for Integrals If f is continuous on [a, b], then a number c in the open interval (a, b) inscribed rectangle Mean Value rect. rectangle area is equal to actual area under curve. Circumscribed Rect a b

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Find the value c guaranteed by the Mean Value Theorem for Integrals for the function f(x) = x 3 over [0, 2]. 4 = 2c 3 c 3 = 2 8 2

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Average Value If f is continuous on [a, b], then the average value of f on this interval is given by Find the average value of f(x) = 3x 2 – 2x on [1, 4]. 40 (1,1) Ave. = 16 16

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The Second Fundamental Theorem of Calculus If f is continuous on an open interval I containing a, then for every x in the interval Ex. Apply the Second Fund. Thm. of Calculus

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Find the derivative of F(x) = Apply the second Fundamental Theorem of Calculus with the Chain Rule.

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