009B Sample Midterm 3, Problem 1 Detailed Solution

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

ExpandBackground Information:
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:

ExpandStep 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  $[0,\pi ]$  is
${\frac {\pi }{4}}{\bigg (}f{\bigg (}{\frac {\pi }{4}}{\bigg )}+f{\bigg (}{\frac {\pi }{2}}{\bigg )}+f{\bigg (}{\frac {3\pi }{4}}{\bigg )}+f(\pi ){\bigg )}.$

ExpandStep 2:
Thus, the right-endpoint Riemann sum is
${\begin{array}{rcl}\displaystyle {{\frac {\pi }{4}}{\bigg (}\sin {\bigg (}{\frac {\pi }{4}}{\bigg )}+\sin {\bigg (}{\frac {\pi }{2}}{\bigg )}+\sin {\bigg (}{\frac {3\pi }{4}}{\bigg )}+\sin(\pi ){\bigg )}}&=&\displaystyle {{\frac {\pi }{4}}{\bigg (}{\frac {\sqrt {2}}{2}}+1+{\frac {\sqrt {2}}{2}}+0{\bigg )}}\\&&\\&=&\displaystyle {{\frac {\pi }{4}}({\sqrt {2}}+1).}\\\end{array}}$

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

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