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 two curves cross inside the interval, the top curve stops being the top curve. A single integral across that crossing does not just lose accuracy: the pieces cancel, and rechecking the arithmetic will never find it, because the arithmetic is right. Twelve short steps; press Next for each one.

12 STEPS · 6 QUICK CHECKS · SPLIT AT EVERY INTERIOR ROOT, THEN RE-ORDER EACH PIECE · v2

Split at every interior crossing, then put the top curve first on each piece

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 color 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?

Step 1 / 12