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integration 137Figure 6.4 Throwing stones in a pond as a technique for measuring its area. There is a tutorialon this on the CD where you can see the actual “splashes” (the dark spots) used in anintegration.4. For i =1to N, pick (x i ,y i )=(r 2i−1 ,r 2i ).5. If x 2 i + y2 i < 1, let N pond = N pond +1; otherwise let N box = N box +1.6. Use (6.44) to calculate the area, and in this way π.7. Increase N until you get π to three significant figures (we don’t ask much —that’s only slide-rule accuracy).6.5.2 Integration by Mean Value (Math)The standard Monte Carlo technique for integration is based on the mean valuetheorem (presumably familiar from elementary calculus):I =∫ badx f(x)=(b − a)〈f〉. (6.45)The theorem states the obvious if you think of integrals as areas: The value of theintegral of some function f(x) between a and b equals the length of the interval(b − a) times the mean value of the function over that interval 〈f〉 (Figure 6.5). Theintegration algorithm uses Monte Carlo techniques to evaluate the mean in (6.45).With a sequence a ≤ x i ≤ b of N uniform random numbers, we want to determinethe sample mean by sampling the function f(x) at these points:〈f〉≃ 1 NN∑f(x i ). (6.46)i=1−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 137