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M2-Ch5-9 - M2 M2 Calculus Fall 2011 Integration Ch5 y = f x...

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M2 M2 Calculus Fall 2011
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Integration Ch5 1 3 1 ) ( x f y = dx x f A b a ) ( = < < = elsewhere x x f 0 3 1 1 ) ( 2 ) 2 )( 1 ( 1 3 1 = = = dx A 2 1 3 1 3 1 3 1 = - = = = = = x x x dx A
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Integration Ch5 3 3 ) ( x f y = dx x f A b a ) ( = < < = elsewhere x x x f 0 3 0 ) ( 5 . 4 ) 3 )( 3 )( 5 . 0 ( 3 0 = = = dx x A 5 . 4 2 9 0 2 1 3 2 1 2 1 2 2 3 0 2 3 0 = = - = = = = = x x x dx x A
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Integration Ch5 2 4 ) ( x f y = dx x f A b a ) ( = < < = elsewhere x x x f 0 2 0 ) ( 2 Estimate dx x A ? 2 0 2 = = x xx 0 0 0.5 0.25 1 1 1.5 2.25 2 4 x ) ( x f Area upper estimate Area lower estimate 75 . 3 4 15 ) 4 25 . 2 1 25 . 0 )( 5 . 0 ( = = + + + 75 . 1 4 7 ) 25 . 2 1 25 . 0 0 )( 5 . 0 ( = = + + + ) ( p U f ) ( p L f 2 1 = x 667 . 2 3 8 0 3 1 2 3 1 3 1 2 3 2 0 3 2 0 2 = = - = = = = = x x x dx x A
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Definite Integral Properties Ch5 The definite integral: Theorem: if f is continuous on [a,b], then: Example dx x f b a ) ( dx x f dx x f dx x f b a b c c a ) ( ) ( ) ( = + dx x f dx x f dx x f ) ( ) ( ) ( 13 3 13 8 8 3 = + dx x f dx x f a b b a ) ( ) ( - = 0 ) ( = dx x f b b b c a < <
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The Fundamental Theorem of integral calculus Ch5 The Antiderivative: Theorem: if f is continuous on [a,b], and G is its antiderivative then : Example ) ( ) ( ) ( a G b G dx x f b a - = ) ( ) ( x f x G dx d = 3 7 1 3 1 2 3 1 3 1 2 3 2 1 3 2 1 2 = - = = = = x x x dx x 2 1 ) 1 ( )) 0 cos( ( cos cos sin 0 0 = + - - = - - - = - = = = π π π x x x dx x + = + b b b b b b dx x f dx x g dx x f x
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