Example 1: Integral as Accumulation Tool

Suppose that in a memory experiment the rate of memorization is given by the equation

$M^{\prime }\left(t\right)=–.009t^2+.2t$

where $M^{\prime }\left(t\right)$ is the memory rate in words per minute. How many words are memorized in the first 10-minutes?

Step 1: Check whether you are being asked to find the Net or Total amount.

In a problem like this you want to know the Net amount memorized in the first 10 minutes because you could remember words (positive area) and you could also forget words (negative area). At the end of 10 minutes, they want to know what you memorized, the Net amount.

Step 2: Check the units “on the top” your given rate of change.

Everything checks out, you are given a rate of change in words per minute , and you are being asked to find the amount of words you could memorize in the first 10 minutes.

Given rate of change: words per minute , $\frac{\mathit{words}}{\mathit{minute}}$

Being asked to find, “How many words ?”

Step 3: Check the units “on the bottom” your given rate of change.

Here your units do match up. The rate of change is in terms of words per minute , and the time interval you are being asked about is also in minutes .

Given rate of change: words per minute , $\frac{\mathit{words}}{\mathit{minute}}$

Given time interval: “first 10- minutes ”

Step 4: Setup and solve your Definite Integral.

Here you are given a time interval of the “ first 10-minutes. ” You will want to use this time interval for your upper and lower bounds. The “ first 10-minutes, ” means time starts at $t=0$ and ends at $t=10,\mathit{ or }\left[0, 10\right]$.

You are being asked to find, “how many words ,” which means you are finding how many words you accumulate , Net words, in the given time interval. You will want to use the given words per minute rate of change equation,
$M^{\prime }\left(t\right)=–.009t^2+.2t$.

At the end of it all, DON’T FORGET THE UNITS . Word problems should always have units .

$\underset{0}{\overset{10}{\int }}–.009t^2+.2t\mathit{dt}=$

Power Rule :

${\frac{–.009t^3}{3}+\frac{.2t^2}{2}|}_0^{10}\mathit{=}$

Top minus Bottom :

$\left[\frac{–.009{\left(10\right)}^3}{3}+\frac{.2{\left(10\right)}^2}{2}\right]–\left[\frac{–.009{\left(0\right)}^3}{3}+\frac{.2{\left(0\right)}^2}{2}\right]=$

$\left[7\right]–\left[0\right]=7\mathit{ words}$

Final Result:

At a rate of memorization of $M^{\prime }\left(t\right)=–.009t^2+.2t$a person would be able to memorize 7-words in the first 10-minutes.