Fundamental Theorem of Calculus (Concept)

The Fundamental Theorem of Calculus (FTC) is one of the most profound results in all of mathematics. It establishes a deep and surprising connection between two operations that appear, at first glance, to be completely unrelated: differentiation (finding rates of change) and integration (finding accumulated totals). Before its discovery in the 17th century, these were studied as separate branches of mathematics. The FTC revealed that they are, in fact, inverse processes of one another — just as multiplication and division are inverses. The theorem comes in two parts. Part 1 states that if you define a function as a definite integral with a variable upper limit, then differentiating that function gives back the original integrand. In other words, integration 'undoes' differentiation. Part 2 — the version most commonly used in calculations — states that a definite integral of a function can be evaluated exactly by finding any antiderivative of that function and computing the difference of its values at the endpoints. This transforms integration from a laborious limit-of-sums process into a straightforward algebraic computation. Understanding the FTC unlocks the full power of calculus. It explains why antiderivatives are useful, why the area under a curve is related to slopes, and it forms the theoretical backbone for computing areas, volumes, work, probability, and countless other accumulated quantities in science and engineering. Mastering this concept is essential for every student of calculus.

Key takeaways
  • Setting the Scene: The Accumulation Function
  • Part 1: Differentiating an Integral Recovers the Integrand
  • Part 2: Evaluating Definite Integrals with Antiderivatives
  • Why 'Inverse Processes'?
  • Using the Chain Rule with FTC Part 1

Setting the Scene: The Accumulation Function

Let f be a continuous function on an interval [a, b]. Define a new function A(x) by the rule A(x) = ∫ from a to x of f(t) dt. This function A(x) measures the net signed area accumulated under the curve y = f(t) from the fixed left endpoint a up to the variable point x. Think of it as a running total: as x increases, A(x) records how much area has been swept out so far.

Part 1: Differentiating an Integral Recovers the Integrand

FTC Part 1 states: If f is continuous on [a, b], then A(x) = ∫ from a to x of f(t) dt is differentiable on (a, b), and A'(x) = f(x). Intuitively, the rate at which area accumulates at the point x equals the height of the function at x. To see why: A(x + h) − A(x) ≈ f(x) · h for small h (a thin rectangle of height f(x) and width h), so [A(x+h) − A(x)] / h ≈ f(x), and taking h → 0 gives A'(x) = f(x) exactly.

Part 2: Evaluating Definite Integrals with Antiderivatives

FTC Part 2 (the Evaluation Theorem) states: If F is any antiderivative of f on [a, b] — meaning F'(x) = f(x) — then ∫ from a to b of f(x) dx = F(b) − F(a). This is written using the notation [F(x)] from a to b. The proof follows directly from Part 1: the accumulation function A(x) is an antiderivative of f, and A(a) = 0, so ∫ from a to b of f(x) dx = A(b) = A(b) − A(a) = F(b) − F(a) for any antiderivative F (since all antiderivatives differ by a constant that cancels).

Why 'Inverse Processes'?

Part 1 shows that d/dx [∫ from a to x f(t) dt] = f(x): differentiation undoes integration. Part 2 shows that ∫ from a to b F'(x) dx = F(b) − F(a): integration undoes differentiation (up to a constant). Together these mean that ∫ and d/dx are inverse operations, just like √ and squaring. This symmetry is the heart of the FTC and explains why antidifferentiation — running differentiation rules in reverse — is the key technique for evaluating integrals.

Using the Chain Rule with FTC Part 1

When the upper limit is not simply x but a function g(x), the Chain Rule extends FTC Part 1: d/dx [∫ from a to g(x) f(t) dt] = f(g(x)) · g'(x). For example, d/dx [∫ from 1 to x³ cos(t) dt] = cos(x³) · 3x². This extension is frequently tested and is essential for more advanced calculus topics such as differential equations and integral transforms.

The Key Equation

∫abf(x) dx=F(b)−F(a),where F′(x)=f(x)\int_{a}^{b} f(x)\, dx = F(b) - F(a), \quad \text{where } F'(x) = f(x)
  • f(x) — Integrand (varies): The continuous function being integrated; represents the rate or height whose accumulation is being measured.
  • F(x) — Antiderivative (varies): Any function satisfying F'(x) = f(x). The choice of antiderivative does not matter because the constant of integration cancels in F(b) − F(a).
  • a,\, b — Limits of integration (dimensionless (or same units as x)): The left (lower) and right (upper) endpoints of the interval over which f is integrated.
  • \int_{a}^{b} f(x)\,dx — Definite integral (units of f × units of x): The net signed area between the graph of f and the x-axis from x = a to x = b.

Worked Example

Problem. Evaluate the definite integral ∫ from 1 to 4 of (3x² − 2x + 5) dx using the Fundamental Theorem of Calculus.

∫14(3x2−2x+5) dx=[x3−x2+5x]14=68−5=63\int_{1}^{4}(3x^2 - 2x + 5)\,dx = \left[x^3 - x^2 + 5x\right]_{1}^{4} = 68 - 5 = 63

Answer. 63

Figure
Diagram
Fundamental Theorem of Calculusxyy = f(x)abArea = F(b) − F(a)f(b)f(a)Net signed area under f(x)∫ f(x) dx = F(b) − F(a)∫ba
Origins · 1 minGottfried Wilhelm Leibniz and how this was discovered
GW
Gottfried Wilhelm Leibniz
1646–1716 · German

Leibniz independently co-discovered calculus alongside Isaac Newton and developed the elegant notation we still use today — including the integral sign ∫ (an elongated S for 'summa', meaning sum) and the differential notation dy/dx. He was the first to clearly articulate and publish the Fundamental Theorem of Calculus in 1675–1686, framing integration and differentiation as inverse operations. His notation made calculus far more practical and teachable than Newton's alternative notation. Leibniz chose the elongated 'S' symbol for the integral because he thought of integration as an infinite sum of infinitely thin rectangles. That intuition, encoded in a single symbol, has guided calculus students for over 300 years.

  • Definite Integrals — The FTC Part 2 is the primary tool used to evaluate definite integrals; understanding what a definite integral represents as a Riemann sum gives meaning to what the FTC is computing.
  • Antiderivatives and Indefinite Integrals — FTC Part 2 requires finding an antiderivative F such that F'(x) = f(x); fluency with antidifferentiation rules is essential for applying the theorem.
  • Chain Rule — Differentiating accumulation functions with composite upper limits (e.g., ∫ from a to g(x) f(t) dt) requires combining FTC Part 1 with the Chain Rule.
  • Riemann Sums — Riemann sums provide the limit definition of the definite integral; the FTC explains why the tedious limit process can be replaced by antiderivative evaluation.
  • Net Change Theorem — The Net Change Theorem is a direct application of FTC Part 2, interpreting ∫f(x) dx as the total net change in a quantity whose rate of change is f(x).