009B Sample Final 3, Problem 1

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

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

Step 2:
Thus, the left-endpoint Riemann sum is
${\begin{array}{rcl}\displaystyle {S}&=&\displaystyle {{\frac {1}{2}}{\bigg (}0+{\frac {3}{4}}+1+{\frac {3}{4}}{\bigg )}}\\&&\\&=&\displaystyle {{\frac {5}{4}}.}\end{array}}$

Final Answer:
${\frac {5}{4}}$

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