Areas Under Kinematics Graphs

Displacement is velocity times time, but only when velocity stays constant. When velocity changes, you can't just multiply once. Instead, slice the motion into tiny time intervals so short that velocity is basically constant during each one. Each slice contributes a sliver of displacement equal to velocity times that tiny time, which is the area of a thin rectangle under the curve. Add up every sliver and you get the total displacement, which is exactly the total area under the velocity-time graph. Shrink the slices to zero width and that sum becomes an integral. The same logic applies one level up: acceleration times a tiny time gives a tiny change in velocity, so the area under an acceleration-time graph gives total change in velocity.

This is the flip side of the derivative relationships in this unit. Slope gives you the rate of change going forward in time; area undoes that process, recovering the total change from the rate. Velocity is the derivative of position, so area under velocity gives you displacement. Acceleration is the derivative of velocity, so area under acceleration gives you change in velocity. Graphs made of straight lines let you compute these areas with plain geometry: triangles for ramps, rectangles for constant sections, trapezoids for combinations. Once velocity or acceleration curves instead of staying linear, you switch to actual integration, but the idea is identical — area still means accumulated change.