Calculus (3rd Edition)

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

Chapter 5 - The Integral - 5.1 Approximating and Computing Area - Exercises - Page 235: 21



Work Step by Step

Given $$ f(x)=f(x)=\cos ^{2} x, \quad\left[\frac{\pi}{6}, \frac{\pi}{2}\right]$$ Since $n=4$, $\Delta x= \dfrac{b-a}{n}=\dfrac{\pi}{12}$ and $$x_0= \pi/6,\ x_1= \pi/4,\ x_2= \pi/3,\ x_3=5\pi/12,\ x_4= \pi/2$$ Then \begin{align*} L_{n}&=\left[f(x_0)+f(x_1)+.......+f(x_{n-1})\right]\Delta x\\ L_4&=\left[f(x_0)+f(x_1)+.......+f(x_{5})\right]\Delta x\\ &=\left[ f(\pi/6)+ f( \pi/4)+ f(\pi /3)+f( 5\pi/12) \right]\frac{\pi}{12}\\ &= \left[ \cos ^{2}(\pi/6)+ \cos ^{2}( \pi/4)+ \cos ^{2}(\pi /3)+\cos ^{2}( 5\pi/12) \right]\frac{\pi}{12}\\ &=\approx 0.410 \end{align*}
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.