009B Sample Final 1, Problem 1

Suppose the speed of a bee is given in the table.

Time (s)	Speed (cm/s)
$0.0$	$125.0$
$2.0$	$118.0$
$4.0$	$116.0$
$6.0$	$112.0$
$8.0$	$120.0$
$10.0$	$113.0$

(a) Using the given measurements, find the left-hand estimate for the distance the bee moved during this experiment.

(b) Using the given measurements, find the midpoint estimate for the distance the bee moved during this experiment.

Foundations:
1. The height of each rectangle in the left-hand Riemann sum is given by choosing
the left endpoints of each interval.
3. The height of each rectangle in the midpoint Riemann sum is given by
${\frac {f(a)+f(b)}{2}}$  where  $a$  is the left endpoint of the interval and  $b$  is the right endpoint of the interval.

Solution:

(a)

Step 1:
To estimate the distance the bee moved during this experiment,
we need to calculate the left-hand Riemann sum over the interval  $[0,10].$
Based on the information given in the table, we will have  $5$  rectangles and
each rectangle will have width  $2.$

Step 2:
Let  $s(t)$  be the speed of the bee during the experiment.
Then, the left-hand Riemann sum is
${\begin{array}{rcl}\displaystyle {2(s(0)+s(2)+s(4)+s(6)+s(8))}&=&\displaystyle {2(125+118+116+112+120)}\\&&\\&=&\displaystyle {1182{\text{ cm}}.}\end{array}}$

(b)

Step 1:
To estimate the distance the bee moved during this experiment,
we need to calculate the Riemann sum using the midpoint rule over the interval  $[0,10].$
Based on the information given in the table, we will have  $5$  rectangles and
each rectangle will have width  $2.$

Step 2:
Let  $s(t)$  be the speed of the bee during the experiment.
Then, the Riemann sum using the midpoint rule is
${\begin{array}{rcl}\displaystyle {2{\bigg (}{\frac {s(0)+s(2)}{2}}+{\frac {s(2)+s(4)}{2}}+{\frac {s(4)+s(6)}{2}}+{\frac {s(6)+s(8)}{2}}+{\frac {s(8)+s(10)}{2}}{\bigg )}}&=&\displaystyle {1170{\text{ cm}}.}\end{array}}$

Final Answer:
(a)    $1182{\text{ cm}}$
(b)    $1170{\text{ cm}}$

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