Fundamental Theorem

Level 1 - Math I (Physics) topic page in Integration.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) establishes the profound connection between differentiation and integration - two seemingly opposite operations.

Part 1: The Integral as an Antiderivative

If \(f\) is continuous on \([a,b]\), define:

FTC Part 1

\[F(x) = \int_a^x f(t) \, dt\]

Then \(F\) is differentiable and \(F'(x) = f(x)\).

This tells us that integration can be reversed by differentiation.

Derivative of Integral

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

Part 2: Evaluation Theorem

If \(f\) is continuous on \([a,b]\) and \(F\) is any antiderivative of \(f\), then:

FTC Part 2

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

This provides a practical method to evaluate definite integrals.

Why This Matters

The FTC allows us to:

Compute definite integrals without using Riemann sums
Understand that differentiation and integration are inverse operations
Solve real-world problems involving accumulation

Example Using FTC Part 2

Evaluate \(\int_1^4 (2x + 1) \, dx\):

Example

\[\int (2x + 1) \, dx = x^2 + x + C\]

Evaluate

\[\int_1^4 (2x + 1) \, dx = [x^2 + x]_1^4 = (16 + 4) - (1 + 1) = 20 - 2 = 18\]

Connecting Derivatives and Integrals

The two parts of the FTC show:

Connection

\[\frac{d}{dx}\int_a^x f(t) \, dt = f(x) \quad \text{and} \quad \int_a^b F'(x) \, dx = F(b) - F(a)\]

This duality is central to mathematical analysis and enables solving problems in physics, engineering, and economics involving rates of change and accumulated quantities.

Antiderivatives

Reverse Differentiation