Understanding Integrals as Inverses of Derivatives: Area Under Velocity Graphs and the Fundamental Theorem of Calculus

Overview

In this video, 3Blue1Brown explains how integrals are inverses of derivatives by modeling a car's motion. By looking at velocity as a function of time, the distance traveled is the area under the velocity time curve, which leads to the antiderivative and the fundamental theorem of calculus.

• The link between velocity and distance via the area under a velocity-time graph
• How summing small rectangle areas approximates the distance traveled
• Antiderivatives and how the distance function is an area function
• The fundamental theorem of calculus as a bridge between integration and differentiation

Overview

The video revisits the idea that integrals represent accumulation and that they serve as the inverse operation to differentiation. Using a moving car as a concrete example, the velocity is modeled by a simple function V(t) and the challenge is to recover the distance traveled from velocity data alone. This sets the stage for viewing distance as the area under the velocity curve, a perspective that both intuition and formalism support.

The moving car intuition

Suppose your speed is represented by a velocity function V(t) over a time interval, for instance from t = 0 to t = 8 seconds. If velocity were constant, distance would be simply velocity times time. When velocity varies, you can still think of distance as the sum of many tiny contributions during each short time step. This leads to the idea of approximating the motion by constant velocity on very small intervals and then refining the interval size. The height of each rectangle corresponds to the velocity at the start of the interval, and the width is the time step. Summing the areas of all these rectangles approximates the total distance traveled, and taking the limit as the width goes to zero yields the exact distance equal to the area under the velocity curve.

From approximation to integral

The expression for the accumulated distance is written as an integral of the velocity function between the lower and upper time bounds. This integral is the precise distance traveled as a function of the upper bound t, denoted S(t). The distance function S is thus an antiderivative of V, up to an additive constant. Because the distance traveled from time 0 to 0 is zero, the constant is fixed so that S(0) = 0 in this example. This yields a concrete antiderivative for the chosen velocity function, which can be evaluated to obtain the distance traveled over any interval.

Antiderivatives and the fundamental theorem

Expanding the velocity function to see what integrates to it makes the idea clear. If V(t) = 8t minus t squared, then an antiderivative is 4t squared minus one third t cubed, plus an arbitrary constant. The lower bound subtraction in definite integrals fixes this constant by ensuring the integral from the lower bound to itself is zero. The fundamental theorem of calculus then states that the integral of f over [a, b] is F(b) minus F(a) where F is any antiderivative of f. In our traffic example with V(t) = t(8 minus t), the distance traveled from 0 to 8 seconds works out to a numerical value, while the signed area interpretation explains what happens when velocity becomes negative along the path.

Signed area and negative velocity

When the velocity function dips below the time axis, the corresponding rectangle areas contribute negatively to the sum. This reflects backward motion and is a natural feature of integrals, which measure signed area rather than geometric area alone. If you want the actual distance between the start and end points, you would subtract the portion corresponding to negative velocity, just as integrals use signed areas in general.

Putting it together and looking ahead

The video closes with a recap of how integrating velocity yields distance, how the area under a curve provides a unifying lens for a broad class of problems, and how the fundamental theorem provides a powerful shortcut from accumulation to a simple two-point evaluation. The approach generalizes beyond the moving car to many context where a rate of change is integrated to obtain an accumulated quantity, a core idea across mathematics and the sciences.