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.
Key Formula
∫[a to b] f(x) dx = F(b) − F(a)
LaTeX: \int_a^b f(x)\, dx = F(b) - F(a)
| Symbol | Meaning | Unit |
|---|---|---|
| f(x) | Integrand | dimensionless |
| F(x) | Antiderivative of f(x) | dimensionless |
| a | Lower limit of integration | dimensionless |
| b | Upper limit of integration | dimensionless |
Worked Example
Problem
Evaluate the definite integral ∫[0 to 3] (2x + 1) dx.
Solution
Step 1: Find the antiderivative: F(x) = x² + x. Step 2: Apply the Fundamental Theorem of Calculus: F(b) − F(a). F(3) = (3)² + (3) = 9 + 3 = 12. F(0) = (0)² + (0) = 0. Step 3: Compute: 12 − 0 = 12.
Answer
∫[0 to 3] (2x + 1) dx = 12
Properties of Definite Integrals
| Property | Formula | Description | Example |
|---|---|---|---|
| Reversal of Limits | ∫[a to b] f dx = −∫[b to a] f dx | Swapping limits negates result | ∫[3 to 0] (2x+1) dx = −12 |
| Zero Width | ∫[a to a] f dx = 0 | Integral over zero-width interval is 0 | ∫[2 to 2] x dx = 0 |
| Additivity | ∫[a to c] = ∫[a to b] + ∫[b to c] | Split at interior point | ∫[0 to 3] = ∫[0 to 1] + ∫[1 to 3] |
| Constant Multiple | ∫k·f dx = k·∫f dx | Constants factor out | ∫5x dx = 5∫x dx |
| Sum Rule | ∫(f+g) dx = ∫f dx + ∫g dx | Integral of sum = sum of integrals | ∫(x+x²) dx |
Interactive Tools
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Related Terms
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.
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.
The word "definite" comes from Latin "definitus" meaning "bounded" or "limited." Leibniz introduced the notation ∫[a to b] in the 17th century to denote integration bounded between two specific values.