Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 8 - Techniques of Integration - 8.9 Numerical Integration - Exercises - Page 457: 8


$$ M_6=0.88,\ \ \ \ T_6=0.883 $$

Work Step by Step

Given $$\int_{0}^{\pi/4} \sec xd x,\ \ \ N=6 $$ Since $\Delta x=\dfrac{b-a}{N}=\dfrac{\pi/4}{6}=\dfrac{\pi}{24} $ Therefore, the sub intervals consist of $$[0,\pi/24],[\pi/24,2\pi/24],[2\pi/24,3\pi/24],[3\pi/24,4\pi/24],[4\pi/24,5\pi/24],[5\pi/24,\pi/4]$$ The midpoints of these sub intervals are $$\{\frac{\pi}{48}, \frac{3 \pi}{48}, \ldots, \frac{11 \pi}{48}\}$$ Hence \begin{align*} M_{n}&= \sum_{i=1}^{n}f(m_i)\Delta x\\ M_{6}&= \left[ f(m_1)+ f(m_2)+ ..............+f(m_5)\right]\Delta x\\ &=\frac{\pi}{12}\left[ f( \pi/48)+ f(3\pi/48)+\cdots + f(11\pi/48)\right]\\ &\approx 0.88 \end{align*} Now find $T_6$ \begin{align*} T_{n}&=\dfrac{1}{2}\left[f(x_0)+2f(x_1)+2f(x_2)+.....+2f(x_{n-1})+f(x_n)\right]\Delta x\\ T_{6}&=\dfrac{1}{2}\left[f(x_0)+2f(x_1)+2f(x_2)+2f(x_{3})+2f(x_4)+2f(x_5)+f(x_6)\right]\Delta x\\ &=\frac{\pi}{48}\left[f(0)+2f(\pi/24)+\cdots +2f(5\pi/24)+f(\pi/4)\right] \\ &\approx 0.883 \end{align*}
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