Difference between revisions of "009B Sample Midterm 3, Problem 1"

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[[File:9B_SM3_1_GP.png|right|375px]]
 
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<span class="exam">Divide the interval &nbsp;<math style="vertical-align: -5px">[0,\pi]</math>&nbsp; into four subintervals of equal length &nbsp; <math>\frac{\pi}{4}</math> &nbsp; and compute the right-endpoint Riemann sum of &nbsp;<math style="vertical-align: -5px">y=\sin (x).</math>
 
<span class="exam">Divide the interval &nbsp;<math style="vertical-align: -5px">[0,\pi]</math>&nbsp; into four subintervals of equal length &nbsp; <math>\frac{\pi}{4}</math> &nbsp; and compute the right-endpoint Riemann sum of &nbsp;<math style="vertical-align: -5px">y=\sin (x).</math>
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[[009B Sample Midterm 3, Problem 1 Solution|'''<u>Solution</u>''']]
  
  
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[[009B Sample Midterm 3, Problem 1 Detailed Solution|'''<u>Detailed Solution</u>''']]
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!Foundations: &nbsp;
 
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'''1.''' The height of each rectangle in the right-hand Riemann sum
 
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|&nbsp; &nbsp; &nbsp; &nbsp; is given by choosing the right endpoint of the interval.
 
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'''2.''' See the Riemann sums (insert link) for more information.
 
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'''Solution:'''
 
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!Step 1: &nbsp;
 
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|Let &nbsp;<math style="vertical-align: -5px">f(x)=\sin(x).</math>  
 
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|Each interval has length &nbsp;<math>\frac{\pi}{4}.</math>
 
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|Therefore, the right-endpoint Riemann sum of &nbsp;<math style="vertical-align: -5px">f(x)</math>&nbsp; on the interval &nbsp;<math style="vertical-align: -5px">[0,\pi]</math>&nbsp; is
 
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&nbsp; &nbsp; &nbsp; &nbsp; <math>\frac{\pi}{4}\bigg(f\bigg(\frac{\pi}{4}\bigg)+f\bigg(\frac{\pi}{2}\bigg)+f\bigg(\frac{3\pi}{4}\bigg)+f(\pi)\bigg).</math>
 
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!Step 2: &nbsp;
 
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|Thus, the right-endpoint Riemann sum is
 
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&nbsp; &nbsp; &nbsp; &nbsp; <math>\begin{array}{rcl}
 
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\displaystyle{\frac{\pi}{4}\bigg(\sin\bigg(\frac{\pi}{4}\bigg)+\sin\bigg(\frac{\pi}{2}\bigg)+\sin\bigg(\frac{3\pi}{4}\bigg)+\sin(\pi)\bigg)} & = & \displaystyle{\frac{\pi}{4}\bigg(\frac{\sqrt{2}}{2}+1+\frac{\sqrt{2}}{2}+0\bigg)}\\
 
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&&\\
 
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& = & \displaystyle{\frac{\pi}{4}(\sqrt{2}+1).}\\
 
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\end{array}</math>
 
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!Final Answer: &nbsp;
 
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|&nbsp;&nbsp; &nbsp; &nbsp; <math>\frac{\pi}{4}(\sqrt{2}+1)</math>
 
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[[009B_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']]
 
[[009B_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 17:32, 12 November 2017

Divide the interval    into four subintervals of equal length     and compute the right-endpoint Riemann sum of  

Solution


Detailed Solution


Return to Sample Exam

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