Fundamental Theorem of Calculus: Interactive Visualization

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Fundamental theorem of calculus in this visualization

The fundamental theorem of calculus connects the derivative and the integral. Its first part says that if $f$ is continuous on an interval containing $a$ , the area function (also called the accumulation function, or an integral with a variable upper limit)

Part 1

A(x) = \int_a^x f(t)\,dt \quad\text{has}\quad A'(x) = f(x).

$A(x)$ is also written $\Phi(x)$ . Its second part follows: if $F$ is any antiderivative of $f$ , then $F - A$ has derivative 0, so it is constant, and

Part 2

\int_a^b f(t)\,dt = F(b) - F(a).

The upper graph is $f$ ; the shaded region runs from the lower limit $a$ to the upper limit x. The lower graph is A, drawn on the same horizontal axis. At $x$ , the blue segment above is the height f(x). Where $f$ is continuous at $x$ , the blue tangent below has exactly that slope: its rise over one unit is the same blue segment, because both graphs use the same scale. Where $f$ jumps, there is no tangent; the two one-sided slopes are drawn dashed. Over the thin strip of width Δx beside $x$ , the area grows by $\Delta A \approx f(x)\,\Delta x$ , so $\Delta A / \Delta x \to f(x)$ : the readout bar shows this quotient, and below, the strip's secant on $A$ , whose slope it is, turns into the tangent as $\Delta x \to 0$ . Where $f$ jumps, a strip on each side keeps its own height however thin it gets.

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Reading it correctly

The area is signed. Where $f < 0$ the region counts negative, so $A$ falls; where $x < a$ the integral runs backward and every sign flips. A zero of $f$ where it changes sign is a peak or a valley of $A$ ; a zero without a sign change is not.
The theorem needs $f$ to be continuous at $x$ . Where $f$ jumps, $A$ stays continuous but has a corner: the one-sided slopes are the one-sided limits of $f$ , and $A'(x)$ does not exist. For the step function here, $A$ is integrated piece by piece; no single antiderivative works across the jump.
Changing $a$ adds a constant: $A(x) = F(x) - F(a)$ . When $f$ is continuous, every area function is an antiderivative, but not every antiderivative is an area function: for $f = \cos$ , $\sin x + 5$ is never $\int_a^x \cos t\,dt$ , because that would need $\sin a = -5$ .
The letter $t$ is only the variable of integration; $x$ is where the area stops. The values on this page come from formulas for the area — $F(x) - F(a)$ , piece by piece for the step function — not from numerical sums. For $e^{-t^2}$ the formula uses the error function $\operatorname{erf}$ .

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Where it is used

Evaluating definite integrals: find any antiderivative and subtract, instead of taking a limit of Riemann sums.
Accumulation: displacement is the integral of velocity, and its rate of change is the velocity itself. (Distance travelled is the integral of the speed $|v|$ .)
Defining new functions: $\ln x = \int_1^x \frac{dt}{t}$ , and $\int_0^x e^{-t^2}\,dt = \frac{\sqrt{\pi}}{2}\operatorname{erf} x$ , which has no formula in elementary functions — the theorem still gives its derivative, $e^{-x^2}$ .

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Controls

Drag across either graph to move $x$ , and the $a$ mark on the upper axis to move the lower limit; under Limits each has a wheel for fine steps and keys for its usual values, which land exactly (π stays π).
Under Thin strip, a logarithmic fader sets the width $\Delta x$ from 1 down to 0.0001 (type it down to 0.000001), and a lever puts the strip right of $x$ , left of it, or on both sides. Δx → 0 makes the strip ten times narrower at each step, pausing at each, and stays at the end; pressed there, it starts again from 0.5.
Presentation mode opens a full-screen view for projectors (Space runs Δx → 0, ↑ ↓ make the strip ten times wider or narrower, ← → move $x$ , H hides the faint rest of $A$ , its tangent and its slope).

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Fundamental theorem of calculus in this visualization

Reading it correctly

Where it is used

Controls

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