Fundamental Theorem of Calculus

Core Concept

The Fundamental Theorem of Calculus (FTC) establishes the inverse relationship between differentiation and integration, the two main operations in calculus. It shows that differentiation "undoes" integration, and integration "undoes" differentiation (up to a constant).

Two Connected Parts

FTC Part 1: The Derivative of an Integral

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

\[ F(x) = \int_a^x f(t)\,dt \quad \text{for } a \le x \le b, \]

then \( F \) is differentiable on \((a, b)\) and

\[ F'(x) = f(x). \]

Meaning: The function that accumulates area under \( f \) from \( a \) to \( x \) has a derivative equal to the value of \( f \) at \( x \).

FTC Part 2: The Integral of a Derivative

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

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

Notation: Often written as \( \displaystyle \Big[ F(x) \Big]_{a}^{b} \) or \( F(x) \big|_{a}^{b} \).

Meaning: To compute a definite integral, find an antiderivative and evaluate its net change over \([a, b]\).

Why "Fundamental"?

Before the FTC, calculating areas (integration) and instantaneous rates (differentiation) were separate problems. This theorem unified them, showing they are inverse operations. This made practical computation of areas possible through antiderivatives.

Concrete Example

Let \( f(x) = 3x^2 \)

Step 1: Using FTC Part 2 to compute \( \displaystyle \int_{1}^{2} 3x^2\,dx \):

An antiderivative is \( F(x) = x^3 \) since \( F'(x) = 3x^2 \).

\[ \int_{1}^{2} 3x^2\,dx = F(2) - F(1) = 2^3 - 1^3 = 8 - 1 = 7. \]

Step 2: Illustrating FTC Part 1:

Define \( G(x) = \int_{1}^{x} 3t^2\,dt \).

By FTC Part 1, \( G'(x) = 3x^2 \), which is \( f(x) \).

We can also compute \( G(x) \) directly: \( G(x) = x^3 - 1 \), and indeed \( G'(x) = 3x^2 \).

Visual Intuition

Imagine driving a car:

• • The speedometer reading is the derivative of position (rate of change).
• • The total distance traveled is the integral of speed over time.

FTC Part 1 says: "The rate at which accumulated distance grows equals current speed."

FTC Part 2 says: "If you know your position function, the distance traveled from time \( a \) to \( b \) is just position(\( b \)) minus position(\( a \))."

Historical Significance

Developed independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, the FTC provided the crucial link that transformed calculus from a collection of techniques into a coherent mathematical framework. It enabled solutions to problems in physics, engineering, and mathematics that were previously intractable.

Summary: The Fundamental Theorem of Calculus connects differential calculus (studying rates of change) with integral calculus (studying accumulation of quantities). It guarantees that every continuous function has an antiderivative and provides a practical method for evaluating definite integrals.

Mathematics notation: \( \int \) = integral sign, \( d/dx \) = derivative, \( F' \) = derivative of \( F \).