How to solve

Phase $1$: Visualizing

Start by making a scatterplot of your data.

Next, we are going to use a Riemann Sum to estimate the integral of the data. That is$,$ we want to find the area between the data $($imagine a smooth curve connecting the points$)$ and the $x-$axis.

Phase $2$: Prepping the integral

Especially when we're doing integrals, we need to be careful about units. Specifically, we need to make sure that the units of the rate of change function match the units of our independent variable $($usually time$).$ We're going to estimate the change by multiplying them together, so for example, if we multiplied velocity in mph by seconds, we wouldn't get a meaningful result:

$\frac{\text{m}\text{i}\text{l}\text{e}\text{s}}{\text{h}\text{o}\text{u}\text{r}}*$ second $=\frac{\text{m}\text{i}\text{l}\text{e}\text{s}\text{.}second}{\text{h}\text{o}\text{u}\text{r}}$

We need the units of time to cancel, so that we're just left with miles $($or whatever$).$ So you will need to decide: what units of time will you use? It's up to you. You will either have to change your units of time to match your rate of change, or change your rate of change values to match your units of time, or maybe both. This won't affect your final answer, just the steps of the calculation.

$2.$ Decide what units you will use to measure time, and write that here $($ex: seconds, minutes, days, months, etc.$)$:

Convert your data to the correct units, using the extra column$($s$)$ in the Excel file as needed. Again, you might need to change the independent variable, the dependent variable, or both! Change the header of the new column$($s$)$ to say what the new units are.

$3.$ The data in your file represents some function of time.

For the lake Powell flow rate, let's call it $r(t)$ for flow rate. It represents the rate of change of the total amount of water entering Lake Powell.

For the drag racer, let's call it $v(t),$ for velocity. It represents the rate of change of the distance the car has traveled.

We're going to write a mathematical statement that represents what we're doing in this project. The statement below has five numbered blanks $-$ copy it the same way into your work. Next, write next to each number what should go in that space to complete the mathematical statement.

Change in $q,(1)={\displaystyle \underset{2}{\overset{3}{}}}(4)d(5)$

$1.$

$2.$

$3.$

$4.$

$5.$

Phase $3$: Estimating the integral

$4.$ Now we're going to estimate the integral in question $5.$ First, let's find $L_{16},$ a left$-$hand Riemann Sum with $16$ subintervals. First, find:

$t=\frac{b-a}{n}$

Set up the calculation that you'll use to find the Riemann Sum. Use your value for $t,$ and function notation. For example, if I were estimating the integral of $f(t)$ from $t=2$ to $t=12$ with $t=0\mathrm{.}5,$ I would write

$L_{16}=f(2)(0\mathrm{.}5)+f(2\mathrm{.}5)(0\mathrm{.}5)+f(3)(0\mathrm{.}5)+$dots

Set up your calculation here:

$4.$ Now plug in the values from your data and show your calculation to find $L_{16}.$ Include units with your answer.

$5.$ Next, find $R_{16}.$ You can jump straight to plugging in function values, you don't need to set up the whole sum again. In other words, redo question $5$ for $R_{16},$ but you don't need to redo question $4.$ Now find the average of $L_{16}$ and $R_{16}.$

$6.$ Briefly, explain why it is not possible to find $M_{16}$ for these data. Then set up the sum to find $M_8,$ as you did in question $5$ for $L_{16}.$

Note: You'll need to find a new $t$ here.

$7.$ Plug in the function values and show your calculation to find $M_8.$ Include units with your answer.

$8.$ You found $4$ estimates of the integral. $L_{16},R_{16},$ the average of L and R$,$ and $M_8.$ Which do you think is the best estimate, and why?

Show your calculation, including units with your answer.

$s=$ seconds, $mph=$ miles per hour

$\backslash$table$[[$Time $(s),$Speed $($mph$),$Time $($new units$),$Speed $($new units$)],[0,0,,],[0\mathrm{.}2,1,,],[0\mathrm{.}4,6,,],[0\mathrm{.}6,12,,],[0\mathrm{.}8,19,,],[1,25,,],[1\mathrm{.}2,29,,],[1\mathrm{.}4,33,,],[1\mathrm{.}6,37,,],[1\mathrm{.}8,40,,],[2,43,,],[2\mathrm{.}2,47,,],[2\mathrm{.}4,50,,],[2\mathrm{.}6,52,,],[2\mathrm{.}8,53,,],[3,55,,],[3\mathrm{.}2,56,,],[,,,]]$