Definite Integrals and Area
A definite integral is a signed accumulation. It equals ordinary geometrical area only when the curve stays on or above the -axis over the whole interval.
Separate signed integral from total area
Integral and area can differ
integral = A₁ − A₂
area = A₁ + A₂
- Above the axis contributes positively.
- Below the axis contributes negatively.
- For total area, split at every crossing and add the positive sizes.
Put the upper curve first
Read one vertical strip
upper curve: y = 3
strip height = 3 − eˣ
lower curve: y = eˣ
A = ∫₀ˡⁿ³ (3 − eˣ) dx
The limits are the left boundary and the intersection.
For a region between two curves, a thin vertical strip has height upper y-value minus lower y-value. Find every intersection first because the upper curve can change there.
Worked example
Area between a line and an exponential curve
On , the line lies above . The curves meet at the upper limit because .
Common mistake
Find crossings before claiming an area
- Solve inside the stated interval.
- Split the integral at each crossing.
- Change negative contributions to positive sizes before adding.
The curve crosses the axis at . Find the total area between the curve and the axis from to .