007B Sample Midterm 3, Problem 1 Detailed Solution

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Divide the interval  $[0,\pi ]$  into four subintervals of equal length   ${\frac {\pi }{4}}$   and compute the right-endpoint Riemann sum of  $y=\sin(x).$

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:

Step 1:
Let  $f(x)=\sin(x).$
Each interval has length  ${\frac {\pi }{4}}.$
Therefore, the right-endpoint Riemann sum of  $f(x)$  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
${\begin{array}{rcl}\displaystyle {{\frac {\pi }{4}}{\bigg (}\sin {\bigg (}{\frac {\pi }{4}}{\bigg )}+\sin {\bigg (}{\frac {\pi }{2}}{\bigg )}+\sin {\bigg (}{\frac {3\pi }{4}}{\bigg )}+\sin(\pi ){\bigg )}}&=&\displaystyle {{\frac {\pi }{4}}{\bigg (}{\frac {\sqrt {2}}{2}}+1+{\frac {\sqrt {2}}{2}}+0{\bigg )}}\\&&\\&=&\displaystyle {{\frac {\pi }{4}}({\sqrt {2}}+1).}\\\end{array}}$

Final Answer:
${\frac {\pi }{4}}({\sqrt {2}}+1)$

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