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

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<span class="exam">Divide the interval <math>[0,\pi]</math> into four subintervals of equal length <math>\frac{\pi}{4}</math> and compute the right-endpoint Riemann sum of <math>y=\sin (x)</math>
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<span class="exam">Divide the interval <math style="vertical-align: -5px">[0,\pi]</math> into four subintervals of equal length <math>\frac{\pi}{4}</math> and compute the right-endpoint Riemann sum of <math style="vertical-align: -5px">y=\sin (x).</math>
  
  
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!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
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|Let <math>f(x)=\sin(x)</math>. Each interval has length <math>\frac{\pi}{4}</math>. So, the right-endpoint Riemann sum of <math>f(x)</math> on the interval <math>[0,\pi]</math> is
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|Let <math style="vertical-align: -5px">f(x)=\sin(x).</math> Each interval has length <math>\frac{\pi}{4}.</math> So, the right-endpoint Riemann sum of <math style="vertical-align: -5px">f(x)</math> on the interval <math style="vertical-align: -5px">[0,\pi]</math> is
 
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|<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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::<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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|Thus, the right-endpoint Riemann sum is  
 
|Thus, the right-endpoint Riemann sum is  
 
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|<math>\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)=\frac{\pi}{4}\bigg(\frac{\sqrt{2}}{2}+1+\frac{\sqrt{2}}{2}+0\bigg)=\frac{\pi}{4}(\sqrt{2}+1)</math>
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::<math>\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)=\frac{\pi}{4}\bigg(\frac{\sqrt{2}}{2}+1+\frac{\sqrt{2}}{2}+0\bigg)=\frac{\pi}{4}(\sqrt{2}+1).</math>
 
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Revision as of 15:31, 29 March 2016

Divide the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0,\pi]} into four subintervals of equal length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\pi}{4}} and compute the right-endpoint Riemann sum of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=\sin (x).}


Foundations:  
Recall:
1. The height of each rectangle in the right-hand Riemann sum is given by choosing the right endpoint of the interval.
2. See the Riemann sums (insert link) for more information.

Solution:

Step 1:  
Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=\sin(x).} Each interval has length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\pi}{4}.} So, the right-endpoint Riemann sum of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} on the interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [0,\pi]} is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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).}
Step 2:  
Thus, the right-endpoint Riemann sum is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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)=\frac{\pi}{4}\bigg(\frac{\sqrt{2}}{2}+1+\frac{\sqrt{2}}{2}+0\bigg)=\frac{\pi}{4}(\sqrt{2}+1).}
Final Answer:  
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\pi}{4}(\sqrt{2}+1)}

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