[trimethylpurine]

That's interesting. To me it seems intuitive. It's a real area that can be drawn like any other. The sign is an operator describing what function to visualize, not a property of the measured area. So thinking of it in that way eliminates any need for the term "negative area."

But, intuition is subjective, so you may need to adjust the terminology to fit the visualization.

[FartyMcFarter]

It is intuitive on some higher level, now that I know about signs, operators, negative numbers and all that. But when talking about visual information (i.e. "visual proof"), it isn't intuitive in that context.

In addition to that, for all I know there could be some pitfalls involved with negative areas which I'm not aware of. Even if there aren't any pitfalls, this isn't immediately obvious to someone who isn't familiar with the concept of negative area.

If I'm willing (or forced) to think in such abstract terms, I would much prefer an algebraic proof to this visual proof.

[jacobolus]

There are no serious pitfalls with oriented areas. Adding them to your arsenal of geometric proof tools will greatly simplify many proofs. Not having such a concept makes ancient geometry books much more complicated than they need to be, often requiring lots of detailed case analysis where the separate cases are essentially the same, just oriented opposite ways.

[ses1984]

Your intuition is irrelevant if they’re mathematically equivalent.