..This unit marks a departure from the path AP Calc has taken thus far. Think about it, to this point, you’ve found the slope of the line tangent
to a curve at a point; you’ve calculated velocity
when given a position function; you’ve determined function maxima
. In other words, you’ve investigated rates of change, rates of change, and some more rates of change
Oh, yeah, I suppose that’s right. So...what’s next?
Well, we’ve seen how to analyze the instantaneous rate of change of a curve—say one that measures bacteria population as a function of time; but suppose that we know the rate function—or how the population changes as a function of time. Can this tell us how much the population it measures changes during a specific time interval?
Wait, so we’re still studying change?
Have you heard the saying the more things change the more things stay the same?
It’s true here, as well—with a twist. The difference is that through differentiation we’re measuring rates of change, while integration—our new process—measures net change.
So differentiation and this integration thing, they’re totally different, then?
While integration is a concept and process distinct from differentiation, these ideas are linked. In fact, their underlying connection unifies the branches of differential
calculus. But we’ll talk about the Fundamental Theorem of Calculus
a bit later on.
For now, takeaway #1 should be that integration helps to measure the net change given a rate function. And takeaway #2? Well, this is a fairly important topic—and the College Board® thinks so, too—because whether you sit the AB or BC exam, integration carries a 17-20% weighting!