Chapter 16 Numerical Integration ( ) b a I f x dx = ∫
Graphical Representation of Integral Integral = area under the curve Use of a grid to approximate an integral
Use of strips to approximate an integral
Numerical Integration: applications Net force against a skyscraper Cross-sectional area and volume flowrate in a river Survey of land area of an irregular lot
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
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 )
0 2 4 6 8 10 12 3 5 7 9 11 13 15 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
Newton-Cotes formulas - based on idea dx x f dx x f I b a n b a ∫ ∫ 2245 = ) ( ) ( 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
f n ( x ) can be linear f n ( x ) can be quadratic
f n ( x ) can also be cubic or other higher-order polynomials
Polynomial can be piecewise over the data
Numerical Integration Newton-Cotes Closed Formulae -- Use both end points Trapezoidal Rule : Linear
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- Spring '09
- Numerical Analysis, dx, Newton–Cotes formulas