Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus establishes the deep connection between differentiation and integration, showing that they are inverse operations. It has two parts: the first part states that if F is the integral function of f, then F is differentiable and F'(x) = f(x); the second part provides a practical method for evaluating definite integrals using antiderivatives. This theorem is arguably the most important result in calculus, unifying two independently developed concepts by Newton and Leibniz.
Key Formula
Part 1: d/dx[∫[a to x] f(t) dt] = f(x) Part 2: ∫[a to b] f(x) dx = F(b) − F(a)
LaTeX: \text{Part 1: } \frac{d}{dx}\int_a^x f(t)\, dt = f(x) \qquad \text{Part 2: } \int_a^b f(x)\, dx = F(b) - F(a)
| Symbol | Meaning | Unit |
|---|---|---|
| f(x) | Continuous function | dimensionless |
| F(x) | Antiderivative of f(x) | dimensionless |
| a, b | Lower and upper limits of integration | dimensionless |
| t | Dummy variable of integration | dimensionless |
Worked Example
Problem
Use the Fundamental Theorem of Calculus to evaluate ∫[1 to 4] √x dx.
Solution
Step 1: Write √x = x^(1/2). Find antiderivative using power rule: F(x) = x^(3/2) / (3/2) = (2/3)x^(3/2). Step 2: Apply Part 2 of FTC: F(b) − F(a). F(4) = (2/3)(4)^(3/2) = (2/3)(8) = 16/3. F(1) = (2/3)(1)^(3/2) = (2/3)(1) = 2/3. Step 3: F(4) − F(1) = 16/3 − 2/3 = 14/3.
Answer
∫[1 to 4] √x dx = 14/3 ≈ 4.667
Two Parts of the Fundamental Theorem of Calculus
| Part | Statement | What It Shows | Key Application | Condition |
|---|---|---|---|---|
| Part 1 (FTC1) | d/dx[∫[a,x] f(t) dt] = f(x) | Differentiation undoes integration | Defining integral functions | f continuous on [a,b] |
| Part 2 (FTC2) | ∫[a,b] f = F(b)−F(a) | Integration via antiderivatives | Evaluating definite integrals | F is antiderivative of f |
| Corollary | Differentiation and integration are inverse operations | Unification of calculus | Theoretical foundation | f integrable |
Interactive Tools
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Related Terms
Definite Integral
A definite integral is an integral evaluated over a specific closed interval [a, b], producing a single numerical value that represents the net signed area between the function's curve and the x-axis over that interval. It is defined as the limit of Riemann sums as the number of subintervals approaches infinity. Definite integrals are used extensively in physics for calculating work, displacement, charge, and probability.
Indefinite Integral
An indefinite integral is the general antiderivative of a function, representing a family of functions that differ only by a constant. Unlike the definite integral, it produces a function rather than a number, and includes an arbitrary constant of integration C that accounts for all possible antiderivatives. Indefinite integrals are the starting point for solving differential equations and computing definite integrals via the Fundamental Theorem of Calculus.
The term "fundamental" reflects its foundational role in calculus. The theorem was independently formulated by Isaac Newton (using fluxions) and Gottfried Wilhelm Leibniz (using differentials and integrals) in the late 17th century. James Gregory and Isaac Barrow contributed earlier versions.