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PP 8.7 - Approximate Integration We want to develop more...

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Approximate Integration
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We want to develop more methods to approximate the value of a definite integral when it cannot be found directly using the Fundamental Theorem of Calculus. Methods known: Left-endpoint approximation Right-endpoint approximation = - = n i i n x x f L 1 1 ) ( = = n i i n x x f R 1 ) ( Midpoint approximation ) ( , ) ( 1 2 1 1 i i i n i i n x x x x x f M + = = - = n a b x x x f dx x f n i i b a - = = , ) ( ) ( : Know 1 *
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Another approximation, called the Trapezoidal Rule, results when averaging the right and left endpoints approximations: [ ] [ ] ) ( ) ( 2 ... ) ( 2 ) ( 2 ) ( 2 ) ( ) ( 1 2 1 0 1 1 1 2 1 2 1 n n n i i n i i n n n x f x f x f x f x f Δx x x f x x f L R T + + + + + = + = + = - = = - Area of a trapezoid = ) ( 2 1 2 1 b b h + Why Trapezoidal Rule?
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) ( ) ( 2 1 1 i i x f b x f b x h = = = - [ ] ) ( ) ( ) ( Area 1 2 1 2 1 2 1 i i x f x f x b b h + = + = -
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Comparing the Methods Under estimate Over estimate
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Over estimate Better estimate?
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Comparison Difference Left endpoint 30.625000 -11.541667 Right endpoint 48.125000 6.458333 Midpoint 41.562500 - 0.104167 Trapezoid 41.875000 0.208333 Best
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