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Problem 21 Medium Difficulty

(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?

Answer

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. $T : n=509, \quad M : n=360, \quad S : n=22$

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Video Transcript

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