Since one can compute an approximate value for p using numerical integration of the given…

Since

one can compute an approximate value for π using numerical integration of the given function.

(a) Use the midpoint, trapezoid, and Simpson composite quadrature rules to compute the approximate value for π in this manner for various step sizes h. Try to characterize the error as a function of h for each rule, and also compare the accuracy of the rules with each other (based on the known value of π). Is there any point beyond which decreasing h yields no further improvement? Why?

(b) Implement Romberg integration and repeat part a using it.

(c) Compute π again by the same method, this time using a library routine for adaptive quadrature and various error tolerances. How reliable is the error estimate it produces? Compare the work required (integrand evaluations and elapsed time) with that for parts a and b. Make a plot analogous to Fig. 8.4 to show graphically where the integrand is sampled by the adaptive routine.

(d) Compute π again by the same method, this time using Monte Carlo integration with various numbers n of sample points. Try to characterize the error as a function of n, and also compare the work required with that for the previous methods. For a suitable random number generator, see Section 13.5.

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