A room can never have an area of −20 m², yet an integral can. Geometric area measures space; the area under a curve measures effect, and an effect has a direction — which is what the sign records.
Tell someone a room has an area of square metres and it is nonsense. Tell them an integral came out as and nobody blinks. Both statements are correct, and the reason is that the two are not measuring the same kind of thing.
Geometric area measures how much space a shape occupies. Space cannot run out below nothing, so that kind of area is never negative.
The area under a curve measures something else: the effect of the function on the system it describes. An effect has a direction as well as a size, and direction is exactly what a sign is for.
Above the axis, water runs in and the level rises. At the crossing point the flow is zero. Below the axis the flow has reversed and water runs out.
Nothing about the lower patch is smaller. It is the same kind of quantity, doing the opposite job, so it enters the total with the opposite sign.
Where the region counts as positive area. Where the region of the same size counts as negative area.
The sign is decided by which side of the axis the curve sits on, never by how large the region is.
If a region above the axis and a region below it have the same size, they contribute nothing between them.
You filled the tank with a certain amount and then took exactly that amount out. The reading is zero because you finished where you started, not because nothing was ever there.
A curve lies entirely below the axis on , and the region it encloses with the axis has size . What is ?
No antiderivative was needed. Knowing which side of the axis the curve sits on was enough to fix the sign.
On a curve encloses a region of size above the axis and a region of size below it. Find the integral, and find the total area.
In the tank picture: litres went in, came back out, the tank is litres fuller — but litres moved through the pipe.