The Fundamental Theorems of CalculusFirst Fundamental Theorem of CalculusSlide 3Area Under a CurveSlide 5Second Fundamental Theorem of CalculusSlide 7Slide 8Slide 9AssignmentThe Fundamental Theorems of CalculusLesson 5.4First Fundamental Theorem of Calculus•Given f is continuous on interval [a, b]F is any function that satisfies F’(x) = f(x)•Then( ) ( ) ( )baf x dx F b F a= -�First Fundamental Theorem of Calculus•The definite integralcan be computed byfinding an antiderivative F on interval [a,b]evaluating at limits a and b and subtracting•Try ( )baf x dx�736x dx�Area Under a Curve•Consider •Area = sin cos on 0,2y x xp� �= +� �� �20sin cosx x dxp+�Area Under a Curve•Find the area under the following function on the interval [1, 4]2( 1)y x x x= + +Second Fundamental Theorem of Calculus•Often useful to think of the following form•We can consider this to be a function in terms of x ( )xaf t dt�( ) ( )xaF x f t dt=�View QuickTime MovieView QuickTime MovieSecond Fundamental Theorem of Calculus•Suppose we aregiven G(x)•What is G’(x)?4( ) (3 5)xG x t dt= -�Second Fundamental Theorem of Calculus•Note that•Then•What about ?( )( ) ( )( ) ( )( ) ( ) ( )xaF x f t dtF x F adF x F a f xdx== -- =�( ) ( )axF x f t dt=�Since this is a constant …Since this is a constant …Second Fundamental Theorem of Calculus•Try this12( )1 3xdtF x dtt=+�( ) ( )( ) ( )so '( ) ( )axF x f t dtF a F xF x f x== -=-�Assignment•Lesson 5.4•Page 308•Exercises 3, 7, 9, 11, 15, 19, 21, 25 31, 33, 35, 37, 39, 41,
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