Fundamental theorem of calculus

The formula

The theorem comes in two parts. They are two sides of one statement: differentiation and integration undo each other.

Part 1 (differentiation undoes integration):

  Let F(x) = ∫[a..x] f(t) dt

  Then  F'(x) = f(x)     for f continuous on [a, b]

Reading: build a running total of f from a up to a moving point x. The rate at which that total grows, right now, is just the value of f at x. Accumulate a signal, differentiate the accumulation, and you get the signal back.

Part 2 (the Newton–Leibniz rule — the one you actually compute with):

  ∫[a..b] f(x) dx = F(b) − F(a)     where F is any antiderivative of f, i.e. F' = f

Reading: the total accumulation of a rate f over [a, b] equals the net change in anything whose slope is f. You never have to sum infinitely many slices. Find one function whose derivative is f, evaluate it at the two ends, subtract.

The two parts together guarantee something an engineer leans on without thinking: every continuous rate signal has a well-defined running total, and that total is a smooth function you can differentiate right back.

Where you meet it

• The test-stand odometer / integrator. A flow meter reports gallons per minute; the totalizer reports gallons. That totalizer is Part 2 running in hardware: it integrates the rate to get the accumulated quantity. Same math turns an accelerometer trace into velocity, then velocity into displacement — the double integration behind every inertial navigation box on a missile or a launch vehicle.

• Charge and energy in a capacitor. Current is the rate of charge: i = C · dv/dt. Flip it with the FTC and the charge that flowed is Q = ∫ i dt, and the stored energy is E = ∫ p dt = ∫ v·i dt. When you size a hold-up capacitor for a power supply, you are integrating the load current over the ride-through window — Part 2, on a whiteboard, at a design review.

• RMS on a scope. The "true RMS" number your Tektronix reports is sqrt( (1/T) ∫[0..T] v(t)² dt ). The scope samples v(t)², accumulates it, and hands you the value — the definite integral is the whole point of the measurement.

• Impulse from a thrust curve. Total impulse of a solid motor is ∫ F(t) dt — the area under the thrust-versus-time trace. On the specific-impulse worksheet that area is the number you divide by propellant weight. The burn is messy and non-analytic, so you integrate the measured curve numerically, but you are still invoking the same theorem: area under a rate equals accumulated quantity.

How it works

The mechanism is the mean value idea. Over a tiny step Δx, the running total F grows by roughly f(x)·Δx — the height of the signal times the width of the slice. Divide by Δx, shrink the step, and the growth rate of the total is the signal height. That is Part 1. Part 2 then falls out because the net change of F across [a, b] is just the sum of all those little f(x)·Δx growths — which is the integral.

The single most useful consequence: you do not have to sum slices to get an exact integral. If you can name an antiderivative, two evaluations and a subtraction give the exact area. ∫[1..4] 3x² dx is not a numerical estimate — it is 4³ − 1³ = 63, dead exact, because x³ differentiates to 3x². (Checked numerically: the quadrature returns 63.0000.)

Where people get burned:

• Continuity is a hypothesis, not decoration. Part 1 gives F' = f because f is continuous. Feed the integrator a signal with a jump — a relay closing, a valve slamming, a step in commanded thrust — and the accumulated total is still fine (it's continuous, integration smooths things), but its derivative has a corner or a jump right there. Differentiating a real integrated signal to recover the rate amplifies exactly that discontinuity, plus every bit of sensor noise. Integration is forgiving; differentiation is not. This is why you filter before you differentiate and why you trust a totalizer more than a numerically-differentiated position trace.

• "Any antiderivative" means net change, and net change can hide cancellation. ∫ v dt of a shaker table over a full cycle is near zero — the table went out and came back. That is correct signed area, not a bug. If you wanted distance traveled, not displacement, you needed ∫ |v| dt. The FTC gives you net accumulation. Know which one your gauge is reporting.

• The +C that drops out. Every antiderivative is unique only up to a constant, because a constant has zero slope. In Part 2 the constant cancels in F(b) − F(a), which is why "any antiderivative" works. But the constant is your initial condition. An INS that integrates acceleration to position is doing Part 2 with F(a) set to where you started — get the initial position or velocity wrong and the whole trajectory is offset. Integration drift in inertial systems is that unknown constant, compounded.

• When the integral is not elementary. Plenty of honest functions — e^(−x²), sin(x)/x — have no antiderivative you can write in closed form. The FTC still holds; you just can't use Part 2 to shortcut it. You fall back to numerical integration (trapezoid, Simpson, Gauss quadrature), which sums slices the way the definition always said. The theorem tells you the answer exists and is smooth; it doesn't promise it's nameable.

The theorem also generalizes well past freshman calculus. Relax continuity and you get the Lebesgue version (F differentiable almost everywhere); push it onto surfaces and volumes and it becomes the generalized Stokes theorem, ∫[M] dω = ∫[∂M] ω, which is the umbrella that Green's, Gauss's divergence, and the curl theorems all live under. Every one of those is the same idea: what happens inside a region is bookkept by what crosses its boundary. ^[1]

History

The credit fight over calculus is famous, but the FTC has a quieter and older origin than either of its usual namesakes. James Gregory, a Scot, published the first recognizable statement and proof of a rudimentary form in his 1668 Geometriae Pars Universalis — arguably the first attempt at a calculus textbook — where he showed, geometrically, that the tangent and area problems were inverse to each other. ^[1][2]

Isaac Barrow, Newton's teacher at Cambridge, gave a more general geometric proof in his Lectiones Geometricae of 1670, in the tangent constructions of his tenth lecture (Proposition 11). ^[3] Barrow had the inverse relationship as a theorem about curves, but it was locked in the language of geometry, not yet a computational tool. ^[1][2]

Then his student took it and ran. Isaac Newton, and independently Gottfried Leibniz, saw that the inverse relationship wasn't a curiosity about curves — it was the engine of a whole method. Newton built the surrounding theory of fluxions; Leibniz systematized it into the symbolic calculus of infinitesimals and gave us the ∫ and d notation and the F(b) − F(a) bookkeeping engineers still write today. Leibniz himself credited the underlying result as "an elegant theorem due to Gregory." ^[3] The second part is still sometimes called the Newton–Leibniz theorem for that reason. ^[1][2]

Related tools

• /tools/capacitor-energy
• /tools/inductor-energy
• /tools/rms-peak
• /tools/specific-impulse
• /tools/power-supply-holdup

Sources

1. https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus
2. https://mathshistory.st-andrews.ac.uk/Chronology/11/
3. M. Nauenberg, "Barrow and Leibniz on the fundamental theorem of the calculus" — https://arxiv.org/abs/1111.6145 (quotes Leibniz via J.M. Child, *The Geometrical Lectures of Isaac Barrow*, 1920, p. 140, and Barrow's *Lectiones Geometricae*, Lecture 10, Prop. 11)