Fundamental Theorem of Calculus — Mathematics Knowledge System

Overview

The Fundamental Theorem of Calculus (FTC) is the bridge connecting differentiation and integration, proving they are inverse operations. It has two parts: Part 1 says the derivative of an integral (with variable upper limit) gives back the original function; Part 2 gives a practical method for evaluating definite integrals using antiderivatives — the technique used in essentially every introductory calculus computation.

Intuition

Imagine filling a bathtub. The RATE water flows in (gallons per minute) is like a derivative — it tells you how fast the water level is changing at any instant. The TOTAL amount of water in the tub at time T is like an integral of that flow rate from 0 to T. The Fundamental Theorem says: if you know the total-water-so-far function, its rate of change (derivative) IS the flow rate function you started with. Filling and totaling are inverse processes — exactly like squaring and square-rooting.

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Formal Definition

Definition

Part 1 (differentiating an integral) and Part 2 (evaluating an integral):

\frac{d}{dx}\int_a^x f(t)\,dt = f(x)

If f is continuous, the derivative of its accumulation function gives back f itself

FTC Part 1

\int_a^b f(x)\,dx = F(b) - F(a)

where F is any antiderivative of f (i.e. F' = f)

FTC Part 2

Notation

Notation	Meaning
$F(x)\Big\|_a^b = F(b)-F(a)$	Evaluation bar notation — antiderivative evaluated at the bounds

Proofs

Proof sketch of Part 2, given Part 1

$\text{Let } G(x) = \int_a^x f(t)\,dt$ (Define the accumulation function)
$\text{By Part 1, } G'(x) = f(x)$ (G is itself an antiderivative of f)
$\text{If } F \text{ is any other antiderivative, } F(x) = G(x) + C$ (Two antiderivatives of the same function differ only by a constant)
$F(b) - F(a) = [G(b)+C] - [G(a)+C] = G(b) - G(a) = \int_a^b f(t)\,dt - 0$ (The constants cancel, and G(a)=0 since the integral from a to a is zero)

Properties

Existence of antiderivatives for continuous functions

\text{Part 1 guarantees every continuous function has an antiderivative -- namely its own accumulation function.}

Applications

Given a velocity function, FTC lets you compute total displacement directly via an antiderivative, without summing infinitely many tiny distances by hand.

Worked Examples

Find an antiderivative of 3x²: F(x) = x³.
$F(x) = x^3$
Evaluate F(2) - F(0).
$F(2)-F(0) = 8 - 0 = 8$

Answer: 8

Practice Problems

Difficulty 5/10

Evaluate ∫₀^π sin(x) dx.

Difficulty 5/10

Water flows into a tank at rate r(t) = 2t litres/min. Using the Fundamental Theorem of Calculus, how many litres enter between t = 0 and t = 5 min?

Difficulty 5/10

A beam's bending moment M(x) has dM/dx = V(x), the shear force. If V(x) = 400 − 50x (N) along a beam, find the change in moment from x = 0 to x = 4 m.

Common Mistakes

Common Mistake

Confusing which part of FTC applies — trying to use Part 2's bounds-evaluation on a Part 1-style variable-limit derivative problem.

Part 1 is about differentiating an integral (no bounds evaluation needed — the answer is just the integrand). Part 2 is about evaluating an integral numerically using an antiderivative.

Quiz

According to FTC Part 2, ∫ₐᵇ f(x)dx equals:

f(b) - f(a)F(b) - F(a), where F is an antiderivative of ff'(b) - f'(a)The average of f(a) and f(b)

Flashcards

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Historical Background

James Gregory proved a geometric version of the theorem in 1668. Isaac Barrow, Newton's teacher at Cambridge, proved a more general geometric version around 1670. But it was Newton and Leibniz who recognized the theorem's full power — that it turns integration from a laborious geometric limiting process into simple algebraic antidifferentiation — and built the rest of calculus around that insight.

1668
James Gregory publishes a geometric proof of a special case
James Gregory
c. 1670
Isaac Barrow proves a more general geometric version
Isaac Barrow
1670s-1680s
Newton and Leibniz recognize and exploit the full computational power of the theorem
Isaac Newton, Gottfried Wilhelm Leibniz

Summary

The Fundamental Theorem of Calculus proves differentiation and integration are inverse operations.
Part 1: the derivative of an accumulation function ∫ₐˣf(t)dt returns f(x) itself.
Part 2: ∫ₐᵇf(x)dx = F(b) - F(a) for any antiderivative F — this is how definite integrals are actually computed in practice.
Guarantees every continuous function has an antiderivative.
Turned integration from a laborious limiting process into simple algebra.

References

WebsiteWikipedia — Fundamental theorem of calculus