Fundamental Theorem of Calculus Rigorous: Notes and FAQs

The Fundamental Theorem of Calculus (FTC) is the bridge between differentiation and integration. Part 1 says that differentiating an accumulation function undoes integration: if F(x) = ∫ax f(t) dt with f continuous, then F′(x) = f(x). Part 2 says that integration undoes differentiation: ∫ab f = F(b) − F(a) whenever F′ = f. A rigorous treatment proves both from the definitions, with no hand-waving.

This theorem is why calculus is practical: instead of computing limits of Riemann sums every time, we find one antiderivative and evaluate it twice. Every integral you have ever computed with antiderivatives was an application of Part 2, and the theorem tells us precisely when that shortcut is legal.

Key Concepts

• Accumulation function: F(x) = ∫ax f(t) dt measures the signed area accumulated from a up to x; it is defined for any integrable f.
• FTC Part 1: if f is continuous on [a, b], then F(x) = ∫ax f(t) dt is differentiable on (a, b) and F′(x) = f(x).
• FTC Part 2: if f is continuous on [a, b] and F′ = f on (a, b), then ∫ab f(t) dt = F(b) − F(a).
• Proof idea for Part 1: (F(x+h) − F(x))/h = (1/h)∫xx+h f equals f(ch) for some ch between x and x+h by the mean value theorem for integrals; continuity sends f(ch) → f(x).
• Continuity is essential: if f has a jump discontinuity, the accumulation function can have a corner, and Part 1 fails there.

Formulas

• FTC 1: d/dx ∫ax f(t) dt = f(x), for f continuous. Chain-rule version: d/dx ∫ag(x) f(t) dt = f(g(x))·g′(x).
• FTC 2: ∫ab f(x) dx = F(b) − F(a) where F′ = f. The constant of integration cancels, so any antiderivative works.
• Example: d/dx ∫0x sin(t²) dt = sin(x²) directly from FTC 1, even though sin(t²) has no elementary antiderivative.
• Net change: ∫ab F′(x) dx = F(b) − F(a) reads as "the integral of a rate of change is the net change".

Worked Example

Evaluate ∫14 (2x + 1) dx rigorously using FTC Part 2.

1. Find an antiderivative: F(x) = x² + x works, since F′(x) = 2x + 1. The integrand is continuous on [1, 4], so FTC 2 applies.
2. Evaluate at the endpoints: F(4) = 16 + 4 = 20, and F(1) = 1 + 1 = 2.
3. Subtract: ∫14 (2x + 1) dx = F(4) − F(1) = 20 − 2 = 18.
4. Sanity check by splitting: ∫14 2x dx = [x²]14 = 16 − 1 = 15, and ∫14 1 dx = 3; 15 + 3 = 18. The results agree.

Common Misconceptions

• "The definite integral is defined as F(b) − F(a)." No: the definite integral is defined via Riemann sums, and FTC 2 is a theorem connecting the two — that is the whole point.
• "FTC works for any integrable f." Part 2 needs an antiderivative, and Part 1 needs continuity; a function with a jump discontinuity has an accumulation function with a corner.
• "Every continuous function has an elementary antiderivative." e−x² and sin(x²) are continuous, so FTC 1 gives them antiderivatives, but none expressible with elementary functions.

Real-World Applications

• Physics: recovering position from velocity by integrating, and velocity from acceleration — the net-change form of FTC.
• Probability: computing probabilities from density functions via antiderivatives (cumulative distribution functions).
• Engineering: total energy, charge or fluid flow from rate measurements, all evaluated with FTC 2 instead of Riemann sums.

The Fundamental Theorem of Calculus says differentiation and integration are inverse operations, but rigorously: Part 1 differentiates the accumulation function back to f (needs continuity), and Part 2 evaluates definite integrals through antiderivatives (needs F′ = f). The worked example shows the standard workflow — find any antiderivative, evaluate twice, subtract — and the fine print tells you exactly when that workflow is legal.