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(a) Find the approximations $ T_[10} $, $ M_{10} $, and $ S_{10} $ for $ \displaystyle \int_0^\pi \sin x\ dx $ and the corresponding errors $ E_T $, $ E_M $, and $ E_S $.(b) Compare the actual errors in part (a) with the error estimates given by (3) and (4).(c) How large do we have to choose $ n $ so that the approximations $ T_n $, $ M_n $, and $ S_n $ to the integral in part (a) are accurate to within 0.00001?

a. Trapezoidal rule: $n=83, \quad$ Midpoint rule: $n=59$b. $\left|E_{T}\right| \leq 0.025839 \quad\left|E_{M}\right| \leq 0.012919 \quad\left|E_{S}\right| \leq 0.000170$c. $T : n=509, \quad M : n=360, \quad S : n=22$

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 7

Approximate Integration

Integration Techniques

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

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