009B Sample Final 3, Problem 1

Divide the interval  ${\displaystyle [-1,1]}$  into four subintervals of equal length  ${\displaystyle {\frac {1}{2}}}$  and compute the left-endpoint Riemann sum of  ${\displaystyle y=1-x^{2}.}$

Foundations:
The height of each rectangle in the left-endpoint Riemann sum is given by choosing the left endpoint of the interval.

Solution:

Step 1:
Since our interval is  ${\displaystyle [-1,1]}$  and we are using  ${\displaystyle 4}$  rectangles, each rectangle has width  ${\displaystyle {\frac {1}{2}}.}$
Let  ${\displaystyle f(x)=1-x^{2}.}$
So, the left-endpoint Riemann sum is
${\displaystyle S={\frac {1}{2}}{\bigg (}f(-1)+f{\bigg (}-{\frac {1}{2}}{\bigg )}+f(0)+f{\bigg (}{\frac {1}{2}}{\bigg )}{\bigg )}.}$
Step 2:
Thus, the left-endpoint Riemann sum is

${\displaystyle {\begin{array}{rcl}\displaystyle {S}&=&\displaystyle {{\frac {1}{2}}{\bigg (}0+{\frac {3}{4}}+1+{\frac {3}{4}}{\bigg )}}\\&&\\&=&\displaystyle {{\frac {5}{4}}.}\end{array}}}$

${\displaystyle {\frac {5}{4}}}$