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Problem 20 Medium Difficulty

(a) Find the approximations $ T_{10} $ and $ M_{10} $ for $ \displaystyle \int_1^2 e^{\frac{1}{x}}\ dx $.
(b) Estimate the errors in the approximations of part (a).
(c) How large do we have to choose $ n $ so that the approximations $ T_n $ and $ M_n $ to the integral in part (a) are accurate to within 0.0001?

Answer

a. $T_{10}=2.021976$
$M_{10}=2.019102$
b. $\left|E_{T}\right| \leq .0068$
$\left|E_{M}\right| \leq .0034$
c. Trapezoidal rule: $n=83, \quad$ Midpoint rule: $n=59$

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Video Transcript

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