Method: Integral as Accumulation Tool

The mechanics on an accumulation problem are the same mechanics you would do with a Definite Integral. The real difference is going to come from the fact that you will need to setup the integral entirely on your own based on the language of the problem. This means you will need to check a few details before you just plug your given equation and bounds into the integral, and start solving.

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

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

Make sure that finding the total of the units “on top” will answer your given question.

If your rate of change is talking $\frac{\mathit{miles}}{\mathit{hour}}$, and they are asking you about a total amount of miles , then you are good to go.

If your rate of change is talking $\frac{\mathit{miles}}{\mathit{hour}}$, and they are asking you about a total amount of feet , then you are going to need to do some unit conversion on your rate of change.You would need to convert $\frac{\mathit{miles}}{\mathit{hour}}$ into $\frac{\mathit{feet}}{\mathit{hour}}$ so you can accumulate total amount of feet .

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

Check that the units of your given time interval ( x-interval ) match the units “on the bottom” your given rate of change.

Sometimes they might give you a rate of change as miles per hour , $\frac{\mathit{miles}}{\mathit{hour}}$, and then the time interval given in the problem is in minutes. This means you will need to do a unit conversion either on your time interval units or on your rate of change units.

It really does not matter which units you convert because it is the same amount of time, and what you really care about are the accumulation units “on top”.

I always convert the time interval units because I find changing those units to be the more straightforward process.

Step 4: Setup and solve your Definite Integral.

In most situations you will want to use your time interval as the bounds of your Definite Integral, and your rate of change as the equation you are taking the anti-derivative of.

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