1. Denote I =∫10  (1/(1+4x2)) dx
a. Find the exact value of I (for example, by finding anantiderivative of the integrand).
b. For a generic positive integer n, we partition the interval[0, 1] into n equal subintervals [x0, x1],[x1, x2], . . . , [xn-1,xn]. Denote by Ln, Rn,Mn, Tn the Riemann sums corresponding to left-point,right-point, midpoint and trapezoid rule. Use sigma notation towrite a formula for each Ln, Rn,Mn, Tn.
c. With the help of your calculator, compute L4,R4, M4, T4. Which of them isclosest to I ?
d. Write Matlab codes to compute Ln, Rn,Mn, Tn when n = 8, 16, 32, 64.
e. Denote by en(L)  = |Ln − I | the error term from left-point rule. We usesimilar notations for en(R) ,en(M) , en(T) . It isknown that en(L) ,en(R) ≤ (K(b − a)2)/2n ,en(M) ≤ (K~(b −a)3)/24n2 , en(T) ≤(K~(b − a)3)/12n2 where K =max[a,b] |f'(x)| and K˜ = max[a,b] |f''(x)|.Find n such that the left-point rule gives an error not exceeding e= 0.0001. The same question for the right-point, midpoint,trapezoid rule. Hint: you don’t need to find the exact values of Kand K˜ . An upper bound for each of them would be sufficient forthis problem.
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