009B Sample Final 3, Problem 1 - Revision history

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Created page with "<span class="exam">Divide the interval <math style="vertical-align: -5px">[-1,1]</math> into four subintervals of equal length <math style="vertical-align: -..."

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<span class="exam">Divide the interval  <math style="vertical-align: -5px">[-1,1]</math>  into four subintervals of equal length  <math style="vertical-align: -14px">\frac{1}{2}</math>  and compute the left-endpoint Riemann sum of  <math style="vertical-align: -5px">y=1-x^2.</math>

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!Foundations:  
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|The height of each rectangle in the left-endpoint Riemann sum is given by choosing the left endpoint of the interval.
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'''Solution:'''

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!Step 1:  
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|Since our interval is  <math style="vertical-align: -5px">[-1,1]</math>  and we are using  <math style="vertical-align: -1px">4</math>  rectangles, each rectangle has width  <math style="vertical-align: -13px">\frac{1}{2}.</math>
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|Let  <math style="vertical-align: -6px">f(x)=1-x^2.</math>
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|So, the left-endpoint Riemann sum is
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|        <math style="vertical-align: 0px">S=\frac{1}{2}\bigg(f(-1)+f\bigg(-\frac{1}{2}\bigg)+f(0)+f\bigg(\frac{1}{2}\bigg)\bigg).</math>
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!Step 2:  
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|Thus, the left-endpoint Riemann sum is
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        <math>\begin{array}{rcl}
\displaystyle{S} & = & \displaystyle{\frac{1}{2}\bigg(0+\frac{3}{4}+1+\frac{3}{4}\bigg)}\\
&&\\
& = & \displaystyle{\frac{5}{4}.}
\end{array}</math>
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!Final Answer:  
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|       <math>\frac{5}{4}</math>
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[[009B_Sample_Final_3|'''<u>Return to Sample Exam</u>''']]