Integraci\u00f3n.ppt - Numerical Integration Formulas y f(x Integration M I b lim f x)x f x)dx i max x 0 i 1 i a M A f xi xi I i 1 Graphical Representation

Integraciu00f3n.ppt - Numerical Integration Formulas y f(x...

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Numerical Numerical Integration Formulas Integration Formulas
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max 0 1 1 Integration ( ) ( ) ( ) M b i i a x i M i i i y f(x) I f x x f x dx A f x x I lim  
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Graphical Representation of Integral Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral
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Use of strips to Use of strips to approximate an integral approximate an integral
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Numerical Integration Numerical Integration Net force against a skyscraper Cross-sectional area and volume flowrate in a river Survey of land area of an irregular lot
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Water exerting pressure on the upstream face of a dam: ( a ) side view showing force increasing linearly with depth; ( b ) front view showing width of dam in meters. Pressure Force on a Dam Pressure Force on a Dam p = gh = h
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Integration Integration Weighted sum of functional values at discrete points Newton-Cotes closed or open formulae -- evenly spaced points Approximate the function by Lagrange interpolation polynomial Integration of a simple interpolation polynomial Guassian Quadratures Richardson extrapolation and Romberg integration
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Basic Numerical Integration Basic Numerical Integration Weighted sum of function values ) ( ) ( ) ( ) ( ) ( n n 1 1 0 0 i n 0 i i b a x f c x f c x f c x f c dx x f x 0 x 1 x n x n-1 x f ( x )
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0 2 4 6 8 10 12 3 5 7 9 11 13 15 Numerical Integration Numerical Integration Idea is to do integral in small parts, like the way you first learned integration - a summation Numerical methods just try to make it faster and more accurate
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Newton-Cotes formulas - based on idea ( ) ( ) b b n a a I f x dx f x dx Approximate f ( x ) by a polynomial n n 1 n 1 n 1 0 n x a x a x a a x f ) ( Numerical integration Numerical integration
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f n ( x ) can be linear f n ( x ) can be quadratic
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f n ( x ) can also be cubic or other higher-order polynomials
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Polynomial can be piecewise over the data
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Numerical Integration Numerical Integration Newton-Cotes Closed Formulae -- Use both end points Trapezoidal Rule : Linear Simpson’s 1/3-Rule : Quadratic Simpson’s 3/8-Rule : Cubic Boole’s Rule : Fourth-order* Higher-order methods* Newton-Cotes Open Formulae -- Use only interior points midpoint rule Higher-order methods
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Closed and Open Formulae Closed and Open Formulae (a) End points are known (b) Extrapolation
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Trapezoidal Rule Trapezoidal Rule Straight-line approximation 0 0 1 1 0 0 1 ( ) ( ) ( ) ( ) ( ) ( ) 2 b i i a i f x dx c f x c f x c f x h f x f x 1 x 0 x 1 x f ( x ) L(x)
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Trapezoidal Rule Trapezoidal Rule Lagrange interpolation ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ; , , , ) ( ) ( ) ( b f a f 2 h 2 h b f 2 h a f d h b f d 1 h a f d L h dx x L dx x f b f a f 1 L 1 b x 0 a x a b h h dx d a b a x x b x a let x f x x x x x f x x x x x L 1 0 2 1 0 2 1 0 1 0 1 0 b a b a 1 0 1 0 1 0 0 1 0 1
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