Problem 3 :

solution = 3.4960767390561597472864527865220;

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Problem 4 :

(a)

Exact integral = 2/3

Numerical :

h = 0.01 => Sum1 = 0.6667254718034710

h = 0.005 => Sum2 = 0.6666876731570723

h = 0.0025 => Sum4 = 0.6666741475165486

Order : Order = log_2((Sum1-Sum2)/(Sum2-Sum4))

Numerical Order = 1.482637654940340

Order of convergence : O(h^(3/2))

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(b)

Exact integral = 2

Numerical :

h = 0.01 => Sum1 = 1.939512218968385

h = 0.005 => Sum2 = 1.957227727562510

h = 0.0025 => Sum4 = 1.969755198037164

Order : Order = log_2((Sum1-Sum2)/(Sum2-Sum4))

Numerical Order = 0.4999177467910201

Order of convergence : O(h^(1/2))

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(c)

Riemann Sum :

h = 0.01 => Sum = 17.48111447792693

h = 0.005 => Sum = 25.55044902870799

h = 0.0025 => Sum = 36.96221684679724



Integral from h to 1 :

h = 0.01 => Int = 18

h = 0.005 => Int = 26.28427124746190

h = 0.0025 => Int = 38



Difference : D = Int - Sum

h = 0.01 => D1 = 0.5188855220730666

h = 0.005 => D2 = 0.7338222187539145

h = 0.0025 => D4 = 1.037783153202760



Order : Order = log_2((D1-D2)/(D2-D4))

Numerical Order = -0.4999740993469977

Order = O(h^(-1/2))

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