009B Sample Midterm 3, Problem 1

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:
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 $f(x)=\sin(x).$ Each interval has length ${\frac {\pi }{4}}.$
So, the right-endpoint Riemann sum of $f(x)$ on the interval $[0,\pi ]$ is
${\frac {\pi }{4}}{\bigg (}f{\bigg (}{\frac {\pi }{4}}{\bigg )}+f{\bigg (}{\frac {\pi }{2}}{\bigg )}+f{\bigg (}{\frac {3\pi }{4}}{\bigg )}+f(\pi ){\bigg )}.$

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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