Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - Chapter Review Exercises - Page 461: 101



Work Step by Step

Given $$\int_{\pi / 4}^{\pi / 2} \sqrt{\sin \theta} d \theta$$ We have: $\Delta x=\dfrac{b-a}{n}=\dfrac{\pi}{16}$ Therefore, the midpoints of these sub-intervals are $$\frac{9 \pi}{32}, \frac{11 \pi}{32}, \frac{13 \pi}{32},\frac{15 \pi}{32} $$ Hence \begin{align*} M_{n}&= \sum_{i=1}^{n}f(m_i)\Delta x\\ M_{4}&= \left[ f(m_1)+ f(m_2)+ ..+f(m_4)\right]\Delta x\\ &=\Delta x\left(f\left(\frac{9 \pi}{32}\right)+f\left(\frac{11 \pi}{32}\right)+f\left(\frac{13 \pi}{32}\right)+f\left(\frac{15 \pi}{32}\right)\right)\\ &=\frac{\pi}{16}(\sqrt{\sin \frac{9 \pi}{32}}+\sqrt{\sin \frac{11 \pi}{32}}+\sqrt{\sin \frac{13 \pi}{32}}+\sqrt{\sin \frac{15 \pi}{32}})\\ &\approx 0.744978\approx0.7450 \end{align*}
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