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The Fundamental Theorem of Calculus
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The Fundamental Theorem of Calculus
If 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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Some Notes Provided you can find the antiderivative of f you can now find the definite integral without limits. (YAY!) Notation: You don’t need the constant of integration:
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Examples
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Examples
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Examples
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Examples Find the area of the region bounded by the graph of y=2x2-3x+2, the x-axis, and the vertical lines x=0 and x=2
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The Mean Value Theorem for Integrals
We know The mean value theorem states that somewhere “between” the two there is a rectangle whose area is precisely equal to the area of the region under the curve.
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The Mean Value Theorem for Integrals
If f is continuous on [a,b], then there exists a number c on [a,b] such that f (c) is the average value of f on [a,b]
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Example Note: The area under the graph of f is equal to the area of the rectangle whose height is the average value of f
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