Area between curves: ∫ab [top − bottom] dx, split where they cross

Year 12 Mathematics Advanced · the shading flips colour wherever f − g changes sign · one unsplit integral of f − g is a SIGNED total · the area adds |each piece|
Pair

Graph

f g f on top g on top crossing
0
1

Crossings of this pair: . Set a and b to the outer crossings to enclose the whole region.

On [a, b]
∫ab (f − g) dx, unsplit –
antiderivative of f − g –
area, split at crossings –
crossings inside (a, b) –

Piece by piece

Each piece integrates [top − bottom] for its own sub-interval, so every piece is positive. The unsplit integral adds the pieces with their signs, which is why it can be zero for a region that plainly has area.

Signed or area?

The table above is hidden until you ask for the split. Compare the unsplit integral with the area first: when do they differ?

Which curve is on top?