The Same Method, Lying Down

You'll learnto read a horizontal strip as right minus left, to rewrite every boundary as x in terms of y before integrating, to take the limits from the y-coordinates of the crossings rather than reusing the x ones, and to choose the sweep direction by counting integrals.

Turn the picture ninety degrees and nothing else changes. The strip is horizontal, its length is right minus left, its thickness is dy, and the limits are y-values. That is the whole of it — this is one method with two orientations, not two methods. Two things do have to be done before integrating, and both are quiet enough to skip. Every boundary must be rewritten as x in terms of y. And the limits must be recomputed, because the crossing points that gave the old x-limits give different numbers when you read their other coordinate.

8 STEPS · 6 QUICK CHECKS · REWRITE EVERY BOUNDARY, THEN RE-READ THE LIMITS · v1

A = ∫ (right − left) dy ⇒ every boundary as x = g(y) ⇒ limits are the y-coordinates

Before you start

What you're looking at

The same kind of region as Topic 8.4, sliced the other way. A thin horizontal strip lies across it; its length is the distance between the left and right boundaries and its thickness is dy.

The question

Which boundary is on the right at a given height, between which two heights does the region live, and is this the sweep direction that costs fewer integrals?

Watch for

Both crossing points carry two coordinates. One pair is the limits for a dx integral and the other pair is the limits for a dy integral, and they are never the same two numbers.

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