How does an integral add up signed changes?

A definite integral accumulates contributions across an interval. When the vertical quantity is a rate, a thin contribution has rate times a small interval width. Adding such contributions and taking an appropriate limit gives an accumulated change. Its units multiply: meters per second times seconds produces meters.

The fictional velocity v(t) = t − 1 meters per second rises along a straight line. The time-dependent coefficient has units meters per second squared. Velocity is negative before one second, zero at one second and positive afterward. The initial interval runs from zero to three seconds.

The geometry is exact here. Between zero and one second, the graph forms a triangle below the axis with width one second and height one meter per second. Its signed contribution is negative one half meter. From one to three seconds, the triangle above the axis has width two and height two, giving positive two meters.

The definite integral adds these signed contributions: −0.5 + 2 = 1.5 meters of displacement. An antiderivative, F(t) = t²/2 − t with matching units, gives the same result through F(3) − F(0). This connects the geometric accumulation to symbolic calculus.

Adding the magnitudes instead would give 2.5 meters, the distance traveled in this one-dimensional model. The distinction is essential: motion in opposite directions can cancel in net displacement without erasing the path traveled. A signed integral preserves that directional information rather than treating every region as positive.