One Integral Per Piece

You'll learnwhy a single integral across a crossing subtracts one piece from another, where the splits go, why an absolute value taken at the end repairs nothing, and how to tell a touch from a crossing.

When the curves cross inside the interval, the top curve stops being the top curve. A single integral across that crossing does not merely lose accuracy — the pieces cancel. For y = x³ and y = x it returns exactly zero for a region whose area is one half, and no amount of rechecking the arithmetic will find the error, because the arithmetic is right. The setup is what is wrong, and the moment to catch it is before integrating rather than after.

8 STEPS · 6 QUICK CHECKS · SPLIT AT EVERY INTERIOR ROOT, THEN RE-ORDER EACH PIECE · v1

split at every interior root ⇒ re-order each piece ⇒ ∫|f − g| dx, never |∫ (f − g) dx|

Before you start

What you're looking at

Two curves that meet three times, so they enclose two separate pieces rather than one region. The shading colour records the sign a single integral would give each piece.

The question

How many times do these curves actually meet, where does the interval get split, and which curve is on top on each piece separately?

Watch for

An answer of zero, or one that is smaller than a piece you can see, is a symptom of a missing split rather than of bad arithmetic.

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