|
|
| Line 1: |
Line 1: |
| | <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> | | <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> |
| | + | <hr> |
| | + | (insert picture of handwritten solution) |
| | | | |
| | + | [[009B Sample Midterm 3, Problem 1 Detailed Solution|'''<u>Detailed Solution for this Problem</u>''']] |
| | | | |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Foundations:
| |
| − | |-
| |
| − | |
| |
| − | '''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:'''
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 1:
| |
| − | |-
| |
| − | |Let <math style="vertical-align: -5px">f(x)=\sin(x).</math>
| |
| − | |-
| |
| − | |Each interval has length <math>\frac{\pi}{4}.</math>
| |
| − | |-
| |
| − | |Therefore, 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
| |
| − | |-
| |
| − | |
| |
| − | <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>
| |
| − | |}
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Step 2:
| |
| − | |-
| |
| − | |Thus, the right-endpoint Riemann sum is
| |
| − | |-
| |
| − | |
| |
| − | <math>\begin{array}{rcl}
| |
| − | \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)}\\
| |
| − | &&\\
| |
| − | & = & \displaystyle{\frac{\pi}{4}(\sqrt{2}+1).}\\
| |
| − | \end{array}</math>
| |
| − | |}
| |
| − |
| |
| − |
| |
| − | {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
| |
| − | !Final Answer:
| |
| − | |-
| |
| − | | <math>\frac{\pi}{4}(\sqrt{2}+1)</math>
| |
| − | |}
| |
| | [[009B_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']] | | [[009B_Sample_Midterm_3|'''<u>Return to Sample Exam</u>''']] |
Divide the interval ![{\displaystyle [0,\pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e2a912eda6ef1afe46a81b518fe9da64a832751)
into four subintervals of equal length 
and compute the right-endpoint Riemann sum of 
(insert picture of handwritten solution)
Detailed Solution for this Problem
Return to Sample Exam