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Evaluating Riemann Sum for Arctan Function Using Given Partitions and Test Points

This solution demonstrates the evaluation of the Riemann sum R(P,T,f) for the function f(x) = arctan(x) over the specified partitions P and test points T. The partitions are defined as P = {0, 1/4, 1/3, 1/2, 3/4, 4/5, 1}, and the test points are T = {1/8, 1/3, 5/12, 3/5, 4/5, 7/8}. By substituting these values into the Riemann sum formula, we calculate the area approximation under the curve of the arctangent function, resulting in an estimated value of approximately 0.437.

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Evaluating Riemann Sum for Arctan Function Using Given Partitions and Test Points

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  1. Example 3 Evaluate the Riemann sum R(P,T,f)where f(x) = arctan x, P = {0, 1/4, 1/3, 1/2, 3/4, 4/5, 1}and T = {1/8, 1/3, 5/12, 3/5, 4/5, 7/8}. Solution Substitute into the formula R(P,T,f) = f(t1)(x1-x0) + f(t2)(x2-x1) + f(t3)(x3-x2) + f(t4)(x4-x3) + f(t5)(x5-x4) + f(t6)(x6-x5) where x0=0, x1=1/4, x2=1/3, x3=1/2, x4=3/4, x5=4/5, x6=1 and t1=1/8, t2=1/3, t3=5/12, t4=3/5, t5=4/5, t6=7/8: R(P,T,f) = (arctan 1/8)[1/4 – 0] + (arctan 1/3)[1/3– 1/4] + (arctan 5/12)[1/2-1/3] + (arctan 3/5)[3/4-1/2] + (arctan 4/5)[4/5-3/4] + (arctan 7/8)[1-4/5] • (.124)(.25) + (.322)(.083) + (.395)(.167) + (.540)(.25) + (.675)(.05) + (.719)(.200) • .031+ .027+ .066+ .135 + .034 +.144  0.437

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