3.3 How many subintervals are required to ensure an approximation error of less thanx 10-5 when approximating the integral of cos x on [0, 1] and employing3.3.1 the composite trapezoidal rule?3.3.2 the composite Simpson's rule?

(3.3.1) The composite error of Trapezoidal rule is -112b-an3nf"ξ Where f"ξ= maxa≤x≤bf"x…

Answered: 3.3 How many subintervals are required…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

3.3 How many subintervals are required to ensure an approximation error of less than
x 10-5 when approximating the integral of cos x on [0, 1] and employing
3.3.1 the composite trapezoidal rule?
3.3.2 the composite Simpson's rule?

Expert Solution

Step 1: Finding the number of interval

(3.3.1)

The composite error of Trapezoidal rule is $- \frac{1}{12} {(\frac{b - a}{n})}^{3} n f " (ξ)$

Where $f " (ξ) = \underset{a \leq x \leq b}{m a x} |f " (x)|$

$f (x) = \cos (\frac{π x}{2}) f' (x) = - \frac{π}{2} \sin (\frac{π x}{2}) f " (x) = - {(\frac{π}{2})}^{2} \cos (\frac{π x}{2}) f " (ξ) = \underset{a \leq x \leq b}{m a x} |f " (x)| = {(\frac{π}{2})}^{2}$

$∴$ $\frac{1}{12} {(\frac{b - a}{n})}^{3} n f " (ξ) \leq \frac{1}{2} \times 10^{- 5} \Rightarrow \frac{1}{12} {(\frac{1 - 0}{n})}^{3} n {(\frac{π}{2})}^{2} \leq \frac{1}{2} \times 10^{- 5} \Rightarrow \frac{1}{6 \times 10^{- 5}} {(\frac{π}{2})}^{2} \leq n^{2} \Rightarrow \frac{10^{\frac{5}{2}}}{\sqrt{6}} (\frac{π}{2}) \leq n \Rightarrow 202.87 \leq n ∴ n = 203$