The Area Is the Amount

You'll learnto read an accumulated amount off the area under a rate graph, to get its units by cancelling the two axes, to tell a net change apart from a total area on one picture, and to place the largest amount where the rate crosses zero rather than where the rate is biggest.

A rate times an elapsed time is an amount, and on a graph of the rate that product is an area. Everything in this unit is that one sentence, and everything expensive in it is in the word area — because half of an area can be negative. The same shaded picture answers "how much did the tank change" and "how much water moved" with two different numbers, and nothing in the geometry tells you which one was asked for.

8 STEPS · 6 QUICK CHECKS · A HEIGHT IS A RATE, AN AREA IS AN AMOUNT · v1

height = a rate · width = an elapsed time · area = the accumulated change

Before you start

What you're looking at

Two panels stacked on the same horizontal scale. The upper one is a flow rate in liters per minute against minutes; the shaded region under it is area. The lower one plots how much has been collected by each moment, so the height of the lower curve is the amount of shading standing to its left.

The question

If the height is a rate and the width is a time, what is the area — and what happens to it when the rate goes negative?

Watch for

Area under the axis is drawn in violet and counts negatively. Two lower curves appear in step 4: one subtracts the violet piece and one adds it. They are both correct, and only the wording of a question decides which is wanted.

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