Accumulation

The derivative gives us a mathematical way to talk about something we encounter every day in a hundred different places: the rates at which things change.

The integral lets us describe something equally ubiquitous: accumulation.

You know about accumulation:

• Dust under your bed
  
  
• Cereal poured into your bowl
  
  
• Money in your bank account

Or:

• People on earth
  
  
• Zebra mussels in the Great Lakes
  
  
• Charge on a capacitor in an electronic circuit

All of these things involve changing quantities, and all of them result in accumulations.

If you can provide a clear description of a changing quantity — that is, if you can provide a function — the integral will describe the amount that is accumulating. The integral models accumulations in the same way that the derivative models rates of change.

Calculus wouldn't be so powerful if it didn't describe, with just these two basic models, so much of the changing world.

Like the derivative, the integral is given a careful study in calculus, not precalculus. However, just as precalculus will talk repeatedly about the intuitive notion of "rate", it will also talk repeatedly about the intuitive notion of "accumulation".

Trust your intuition about the meaning of these words, and use your study of precalculus to gain experience with the many places where they are used. When you get to calculus, you'll be pleasantly surprised to find how much you've already mastered.

Before you venture off across the bridge, take a moment to ponder, one last time, that engineering marvel stretching between the peaks of the derivative and the integral: The Fundamental Theorem of Calculus. The derivative is a model of rates of change. The integral is a model of accumulation. The more rapidly something changes, the more quickly it accumulates. The more slowly something changes, the more gradually it accumulates. Is it really so difficult to imagine a connection?

Maybe you can climb the peaks of Calculus, and cross that bridge in the sky, after all...