Fundamental Theorem of Calculus: FTC 1 & 2
The Fundamental Theorem of Calculus (FTC) is the single most important result in all of calculus. It establishes that differentiation and integration are inverse operations — two independently defined processes that perfectly undo each other.
FTC has two parts: Part 1 says that if you define F(x) as the integral of f from a to x, then F'(x) = f(x). Part 2 says that to evaluate ∫[a→b]f(x)dx, find any antiderivative F, then compute F(b) − F(a).
Formula
FTC 1: d/dx[∫[a→x]f(t)dt] = f(x) | FTC 2: ∫[a→b]f(x)dx = F(b)−F(a)
FTC Part 1: The Integral as a Function
Define the accumulation function A(x) = ∫[a→x]f(t)dt — the signed area under f from a to x. FTC Part 1 states: A'(x) = f(x).
Proof intuition: A(x+h) − A(x) = ∫[x→x+h]f(t)dt ≈ f(x)·h for small h (area of thin rectangle). So [A(x+h)−A(x)]/h ≈ f(x). Taking h→0 gives the derivative.
Practical consequence: integration creates antiderivatives. Every continuous function has an antiderivative (even those without closed-form elementary expressions, like e^(−x²)).
With chain rule: d/dx[∫[a→g(x)]f(t)dt] = f(g(x)) · g'(x). Example: d/dx[∫[0→x³]sin(t)dt] = sin(x³) · 3x² = 3x²sin(x³).
FTC Part 2: Evaluating Definite Integrals
FTC Part 2 gives the practical method for exact integration: ∫[a→b]f(x)dx = F(b) − F(a), where F is any antiderivative of f. The constant C cancels.
Notation: [F(x)]ₐᵇ means F(b) − F(a). The vertical bar with limits is standard.
Example: ∫[0→3](x² + 2x)dx = [x³/3 + x²]₀³ = (9 + 9) − (0 + 0) = 18.
Example: ∫[0→π]sin(x)dx = [−cos x]₀π = −cos(π) − (−cos(0)) = 1 + 1 = 2.
The choice of antiderivative does not matter: if F and G are both antiderivatives of f, then G = F + C, so G(b) − G(a) = F(b) + C − F(a) − C = F(b) − F(a). The C always cancels.
| Integral | Antiderivative F(x) | Evaluation F(b)−F(a) | Result |
|---|---|---|---|
| ∫[1→4] 2x dx | x² | 16 − 1 | 15 |
| ∫[0→π/2] cos x dx | sin x | 1 − 0 | 1 |
| ∫[1→e] 1/x dx | ln|x| | ln e − ln 1 | 1 |
| ∫[0→2] eˣ dx | eˣ | e² − e⁰ | e² − 1 ≈ 6.389 |
| ∫[0→1] 3x² dx | x³ | 1 − 0 | 1 |
| ∫[1→2] (x³−x) dx | x⁴/4 − x²/2 | (4−2)−(1/4−1/2) | 9/4 |
Historical Significance and Proof Sketch
Before FTC (Newton and Leibniz, 1670s), differentiation and integration were considered separate disciplines. Tangent problems (derivatives) and area problems (integrals) had been studied independently by Fermat, Barrow, Cavalieri, and others.
FTC unified them into a single coherent theory of calculus. This allowed systematic evaluation of areas that previously required ad hoc geometric arguments.
Proof of FTC Part 2: given FTC Part 1 (A'(x) = f(x)) and any antiderivative F (so F = A + C), then ∫[a→b]f = A(b) − A(a) = A(b) − 0 = F(b) − C − (F(a) − C) = F(b) − F(a). The lower limit a sets A(a) = 0 by definition of A.
Significance: FTC converts the hard problem of computing limits of Riemann sums into the tractable problem of finding antiderivatives. Instead of summing infinitely many infinitesimal rectangles, we evaluate one formula at two points.
Applications of the Fundamental Theorem
Displacement from velocity: if v(t) is velocity, then total displacement = ∫[t₁→t₂]v(t)dt = s(t₂) − s(t₁). FTC connects position (antiderivative of velocity) to displacement (definite integral).
Net change theorem: ∫[a→b]F'(x)dx = F(b) − F(a). The integral of a rate of change equals the net change. Applies to population growth, temperature change, accumulated cost, etc.
Area between curves: area = ∫[a→b][f(x) − g(x)]dx where f ≥ g on [a,b]. Find where the curves intersect (solve f = g) to determine the bounds.
Average value of a function: f_avg = (1/(b−a)) · ∫[a→b]f(x)dx. FTC enables computation of this average by finding the antiderivative.