AP Calculus AB and BC
Area vs Net Signed Area
A definite integral computes net signed area, so region below the horizontal axis counts as negative and can cancel region above it. True geometric area integrates the absolute value of the function, so every piece contributes positively.
Net signed area
Use when: The question asks for the value of the integral, a net change, or a displacement.
True area
Use when: The question asks for the area of a region, or uses the word area geometrically.
Side by side
| Net signed area | True area | |
|---|---|---|
| Integral | ||
| Below the axis counts as | Negative | Positive |
| Can be zero | Yes | Only if is zero throughout |
| Needs sign analysis | No | Yes, split at the zeros |
The parallel with motion is exact. Net signed area is to true area what displacement is to total distance travelled, and both distinctions come from the same place: an integral adds signed quantities.
To compute true area by hand, find where the function crosses the axis, split the integral at those points, and add the absolute values of the pieces.
Area between curves
When finding the area between two curves the integrand is top minus bottom, which is already non-negative, so no absolute value is needed as long as you know which curve is on top.
Frequently asked questions
Does a negative integral mean I made a mistake?
No. A negative value is a perfectly valid net signed area, and it is the right answer whenever more of the region sits below the axis.
In the CED: Unit 6: Integration and Accumulation, Unit 8: Applications of Integration