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Calculating Term by Term

Here is the first calculating strategy: if the resulting expression has many terms added and/or subtracted together, you can calculate the terms one at a time, keeping a running total as you go along. For example, say you mentally evaluate $\int_{2}^{10}\left(u^{3}+2u-3\right)du$, and end up visualizing

It can get complex here, because to evaluate it mentally, there are three groups of information to keep in your head at once:

1. the expression $\frac{u^{4}}{4}+u^{2}-3u$,
2. the limits of integration, 10 and 2, and
3. the running total.

While it's possible to keep track of all of this information mentally, generally speaking most people will find it hard at first, except for simpler expressions. As you practice more, the size of expression that you consider ``simple'' will naturally grow. So what do you do when you are dealing with an expression so large that you have trouble remembering everything? Fortunately, you can often rely on a few things that will help. First, many times when evaluating an integral, the original expression will be written or printed someplace that you can see. In our example, $\int_{2}^{10}\left(u^{3}+2u-3\right)du$ will be physically printed somewhere you can easily glance at. This is not always true; sometimes you will have constructed the integral internally yourself. But if it is available physically, you can use it in a few ways. The limits of integration are in it; you can just glance at the integral twice, when you need each one, rather than remember both of them. Also, glancing at the integrand can jog your memory of what term is next.

The polynomial above is a good example to work with, because it is a mid-size expression. Depending on the size of the expression and your own ability (as aided by the techniques in this chapter), you may be able to hold the whole integrated expression in your memory, and visualize each term as needed. If it seems to be larger than you know how to handle at that point, it may be best to write it down. You can then glance at it as you do the calculation.

To help calculate this, you can utilize a tool called subimages. Using subimages is kind of a cross between windowing and chunking. Imagine you have two cards floating in front of you, stacked over each other. On one of them you will keep a running total, and on the other you will put one of the terms in the expression above. You visualize one of them as having the number 0 on it:

In your mind's eye, bring the bottom one on top, so that you can see it better:

That second one has the first term in the antiderivative. Plug in 10 for u there (the value of the higher limit of integration), and evaluate it.

The other card is a running total, which of course started at zero. Now that you have the value of the first term, switch back to the running total card and add it in:

Then switch back to the other card, which now shows the next term in the series.

Again, plug in for $u=10$, evaluate it, and add it into the running total.

Repeat this process for each of the terms in the expression (in this example, there is only one left, -3u). That done, you run through them again. This time, you (a) use the other limit of integration, and (b) negate each term, multiplying it by -1. So after running through the expression the first time, you would visualize

The top subimage, 2570, is the running total. That's the value of $\frac{1}{4}u^{4}+u^{2}-3u$ at u = 10. The subimage below it, which you can't see well here, contains -30 (the value of the last term, -3u, at u = 10). At this point, we are ready to run through the terms again. Start by putting $-\frac{u^{4}}{4}$ in the subimage for the term, and evaluating it at u = 2:

Do you understand why it is $-\frac{u^{4}}{4}$ instead of $+\frac{u^{4}}{4}$? It is, of course, just because $\int_{a}^{b}f(x)\ dx=F(b)-F(a)$, and we are now evaluating with the lower limit of integration.

Next, you add the value from this term into the subtotal, just as above.

Continue this with the other terms. You finally will end up visualizing

2568 is the final total.