The Odometer and the Speedometer

A car's odometer adds up its speedometer readings over time. The Fundamental Theorem says these are two sides of one coin: distance is the integral of speed, and speed is the derivative of distance.

How it works

The Fundamental Theorem of Calculus

In one sentence

If $f$ is continuous on $[a, b]$ , then $F(x) = \int_a^x f(t)\,dt$ is differentiable with $F'(x) = f(x)$ , and for any antiderivative $G$ of $f$ , $\int_a^b f(x)\,dx = G(b) - G(a)$ .

General formula

{\htmlData{sym=0}{D(t)}} = \int_0^t {\htmlData{sym=1}{v}}({\htmlData{sym=2}{u}})\,d{\htmlData{sym=2}{u}} \quad\Rightarrow\quad {\htmlData{sym=3}{D'(t)}} = {\htmlData{sym=1}{v}}(t), \qquad \int_a^b {\htmlData{sym=1}{v}}(t)\,dt = D(b) - D(a)

The area function grows at the rate of the curve's current height: add a sliver of width $dt$ and height $v(t)$ .

What the symbols mean

$D(t)$
Distance
The odometer: total distance after t minutes.
$v$
Speed
The speedometer reading.
$u$
Earlier time
The dummy variable running from 0 to t.
$D'(t)$
Rate of the odometer
How fast distance accumulates: the speed.

Define $D(t)$ as the area under the speed curve from $0$ to $t$ : the distance so far.
Advancing $t$ a little adds a thin strip of area $\approx v(t)\,\Delta t$ , so $D'(t) = v(t)$ (part 1).
So the distance between two times is $D(b) - D(a)$ , for any antiderivative $D$ of $v$ (part 2).
Integration and differentiation undo each other.

Learn the math

Computing Indefinite IntegralsCalc I · Lesson 41, Integrals The Fundamental Theorem of CalculusCalc I · Lesson 46, Integrals Integrals as SolutionsDiff Eq · Lesson 2, Basic Concepts