Top Minus Bottom, and Which Is Which

You'll learnto build the integrand from a single vertical strip, to get the limits by solving for every root instead of borrowing a number from the prompt, and to decide which curve is on top by substituting rather than by looking.

The height of a vertical strip is top minus bottom, and that single sentence is the whole integrand. Everything hard about this topic happens before you integrate anything. Where does the region start and stop? Which curve is which? Neither question can be answered by looking at the equations, and both have a one-move answer: solve for every root, then substitute a point strictly between two of them. Skip either and the arithmetic that follows is flawless and wrong.

8 STEPS · 6 QUICK CHECKS · SOLVE FOR EVERY ROOT, THEN TEST A POINT · v1

A = ∫ (top − bottom) dx ⇒ limits are x-values ⇒ a negative answer is a reversed subtraction

Before you start

What you're looking at

Two curves and the region trapped between them. A thin vertical strip stands in that region; its height is the distance between the curves and its width is dx, so the area is the sum of the strips.

The question

Where does the region begin and end, and which of the two curves is the upper one across that interval?

Watch for

Every number used as a limit here comes from solving one curve equal to the other. Numbers that merely appear in the wording are not limits, and the picture shows what happens when one is treated as though it were.

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