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:
Recall:
1. The height of each rectangle in the lower Riemann sum is given by choosing
the minimum  $y$  value of the left and right endpoints of the rectangle.
2. The height of each rectangle in the upper Riemann sum is given by choosing
the maximum  $y$  value of the left and right endpoints of the rectangle.
3. The area of the region is given by  $\int _{a}^{b}y~dx$
for appropriate values  $a,b$.

Solution:

(a)

Step 1:
We need to set these two equations equal in order to find the intersection points of these functions.
So, we let  $2(-x^{2}+9)=0$.  Solving for  $x,$  we get  $x=\pm 3$.
This means that we need to calculate the Riemann sums over the interval  $[-3,3]$.

Step 2:
Since the length of our interval is  $6$  and we are using  $3$  rectangles,
each rectangle will have width  $2.$
Thus, the lower Riemann sum is
$2(f(-3)+f(-1)+f(3))\,=\,2(0+16+0)\,=\,32.$

(b)

Step 1:
As in Part (a), the length of our interval is  $6$  and
each rectangle will have width  $2.$ (See Step 1 and 2 for (a))

Step 2:
Thus, the upper Riemann sum is
$2(f(-1)+f(-1)+f(1))\,=\,2(16+16+16)\,=\,96.$

(c)

Step 1:
To find the actual area of the region, we need to calculate
$\int _{-3}^{3}2(-x^{2}+9)~dx.$

Step 2:
We integrate to get
${\begin{array}{rcl}\displaystyle {\int _{-3}^{3}2(-x^{2}+9)~dx}&=&\displaystyle {2{\bigg (}{\frac {-x^{3}}{3}}+9x{\bigg )}{\bigg |}_{-3}^{3}}\\&&\\&=&\displaystyle {2{\bigg (}{\frac {-3^{3}}{3}}+9\times 3{\bigg )}-2{\bigg (}{\frac {-(-3)^{3}}{3}}+9(-3){\bigg )}}\\&&\\&=&\displaystyle {2(-9+27)-2(9-27)}\\&&\\&=&\displaystyle {2(18)-2(-18)}\\&&\\&=&\displaystyle {72}.\\\end{array}}$

Final Answer:
(a)    $32$
(b)    $96$
(c)    $72$

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