Say What the Number Means

You'll learnto get the units from the integrand and the variable, to build an amount from a starting amount plus a change, to keep three different questions about one tank apart, and to write the sentence that earns the interpretation point.

A rate integrated over an interval is an amount of change, in the units of the rate times the units of the variable. Getting the number is the easy half. The half that is actually scored is saying, in a sentence, what that number is — and one tank problem can produce four correct numbers, each of which is the wrong answer to three of the four questions you might be asked about it.

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

gal/min × min = gal ⇒ A(b) = A(a) + ∫ (R − W) dt ⇒ name the quantity, interval and units

Before you start

What you're looking at

One tank, for the whole animation. Water enters at one rate and leaves at another, both in gallons per minute, and 50 gallons are already there when the clock starts.

The question

Which of four things is being asked for — a rate at one instant, how much came in, by how much the amount changed, or how much is there now?

Watch for

A rate is the height of a curve at one moment. An amount is an area under it across an interval. Every confusion in this topic is between those two kinds of object.

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