Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 6 - Review - Exercises: 17

Answer

a) A $ \approx 0.38 $ b) V$ \approx 0.87 $

Work Step by Step

{Step 1 of 4} A) Divide the interval [0,1] into 4 sub-intervals (n=4). The width of each sub-interval $ \Delta x=\frac{1}{4} $ The sub-intervals are $ \left[ 0,\frac{1}{4} \right] , \left[ \frac{1}{4},\frac{1}{2} \right] , \left[ \frac{1}{2},\frac{3}{4} \right] , and \left[ \frac{3}{4},1 \right] . $ Their midpoints are $ \frac{1}{8},\frac{3}{8},\frac{5}{8}, and \frac{7}{8} $. {Step 2 of 4} Use the midpoint rule to estimate the area of region R as A=$ \int _{0}^{1}tan \left( x^{2} \right) dx $ $\approx \frac{1}{4} \left[ tan \left( \left( \frac{1}{8} \right) ^{2} \right) +tan \left( \left( \frac{3}{8} \right) ^{2} \right) +tan \left( \left( \frac{5}{8} \right) ^{2} \right) +tan \left( \left( \frac{7}{8} \right) ^{2} \right) \right] $ $ \approx \frac{1}{4} \left[ 0.015626+0.141559+0.411786+0.961216 \right] $ $ \approx \frac{1}{4} \left[ 1.530187 \right] $ A $ \approx 0.38 $ (up to 2 decimal places) {Step 3 of 4} B) Use the midpoint rule to find the volume of the solid generated by rotating region R about the x-axis. This produces a disk of radius $f(x)=tan(x^2)$ The area of the disk A(x) = $ \pi tan^{2} \left( x^{2} \right) $ {Step 4 of 4} Then volume V $ \approx \sum _{i=1}^{4}A \left( x_{i}^{.} \right) . \Delta x \approx \frac{ \pi }{4} \left[ tan^{2} \left( \left( \frac{1}{8} \right) ^{2} \right) +tan^{2} \left( \left( \frac{3}{8} \right) ^{2} \right) +tan^{2} \left( \left( \frac{5}{8} \right) ^{2} \right) +tan^{2} \left( \left( \frac{7}{8} \right) ^{2} \right) \right] $ $ \approx \frac{ \pi }{4} \left[ 0.000+0.020+0.924 \right] $ $ \approx \frac{ \pi }{4} \left( 1.114 \right) $ V$ \approx 0.87 $
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.