Numerical integration - Numerical Methods for Eng [ENGR...

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Numerical Methods for Eng [ENGR 391] [Lyes KADEM 2007] CHAPTER VI Numerical Integration Topics - Riemann sums - Trapezoidal rule - Simpson’s rule - Richardson’s extrapolation - Gauss quadrature rule Mathematically, integration is just finding the area under a curve from one point to another. It is represented by , where the symbol is an integral sign, the numbers a and b are the lower and upper limits of integration, respectively, the function f is the integrand of the integral, and x is the variable of integration. Figure 1 represents a graphical demonstration of the concept. b a dx x f ) ( Why are we interested in integration: because most equations in physics are differential equations that must be integrated to find the solution(s). Furthermore, some physical quantities can be obtained by integration (example: displacement from velocity). The problem is that sometimes integrating analytically some functions can easily become laborious. For this reason, a wide variety of numerical methods have been developed to find the integral. Figure.6.1 - Integration. I. Riemann Sums Let f be defined on the closed interval [ a , b ], and let be an arbitrary partition of [ a , b ] such as: a = x 0 < x 1 < x 2 < … < x n-1 < x n = b , where x i is the length of the i th subinterval. If c i is any point in the i th subinterval, then the sum = Δ n i i i i i i x c x x c f 1 1 , ) ( Numerical Integration 81
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[ENGR 391] [Lyes KADEM 2007] x i x 1 x 2 ... x i-1 x i x n-1 x n is called a Riemann Sum of the function f for the partition on the interval [ a , b ]. For a given partition , the length of the longest subinterval is called the norm of the partition. It is denoted by || || (the norm of ). The following limit is used to define the definite integral: = Δ = Δ n i i i I x c f 1 0 ) ( lim This limit exists if and only if for any positive number ε , there exists a positive number δ such that for every partition of [ a , b ] with || || < δ , it follows that ε < Δ = n i i i x c f I 1 ) ( for any choice of the numbers c i in the i th subinterval of . If the limit of a Riemann Sum of f exists, then the function f is said to be integrable over [ a , b ] and that the Riemann Sums of f on [ a , b ] approach the number I . = Δ = Δ n i i i I x c f 1 0 ) ( lim , Where = b a dx x f I ) ( Example Find the area of the region between the parabola y = x 2 and the x-axis on the interval [0, 4.5]. Use Riemann’s Sum with four partitions. 2. TRAPEZOIDAL RULE Trapezoidal rule is based on the Newton-Cotes formula that if we approximate the integrand by an n th order polynomial, then the integral of the function is approximated by the integral of that n th order polynomial. 0 , 1
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Numerical integration - Numerical Methods for Eng [ENGR...

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