Area under a curve

Also called Area under the graph, Area under the line

The area under a curve is the region bounded by a plotted line and the horizontal axis over some interval, counted as negative wherever the line dips below the axis. On a physics graph it gives an accumulated quantity such as displacement or impulse.

A slope gives a rate; an area gives a total built up over the interval. Every place AP Physics 1 gives an area a meaning, it is the vertical quantity multiplied by the horizontal one:

Displacement is the area under a velocity-time graph (EK 1.3.A.4.iii).
Change in velocity is the area under an acceleration-time graph (EK 1.3.A.4.iv).
Work is the area under a graph of force as a function of displacement (EK 3.2.A.5).
Impulse is the area under a graph of net force as a function of time (EK 4.2.A.4).
Rotational work is the area under torque against angular position (EK 6.2.A.3).
Angular impulse is the area under torque against time (EK 6.3.B.3 and EK 6.3.C.4).

Signed, always. EK 1.3.A.4.iii spells out that the region meant is the one bounded by the function and the horizontal axis for the interval. Below the axis counts negative and cancels area above it. Add the pieces as unsigned magnitudes and a velocity-time graph hands you distance travelled where the question asked for displacement.

Units are the check. Multiply the axis units. Newtons against seconds gives newton seconds, which is why that area is an impulse rather than a force.

Split the region into rectangles and triangles. Curved boundaries stay estimates in AP Physics 1, since a boundary statement under topic 1.3 keeps nonuniform acceleration qualitative. AP Physics C: Mechanics replaces the estimate with a definite integral, printing $\Delta x = \int v_x(t)\,dt$ and $\Delta v_x = \int a_x(t)\,dt$ on its sheet.

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