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, and the limits are heights. Two quiet things have to happen first, and both are easy to skip. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · REWRITE EVERY BOUNDARY, THEN RE-READ THE LIMITS · v2

A horizontal strip is right minus left, and its limits are heights

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.

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 that costs fewer integrals?

Step 1 / 12