What Is a Definite Integral?
A definite integral is a mathematical tool used to compute the accumulated value of a function over a specific interval. For a function f(x), the definite integral from a to b is written as:
∫ab f(x) dx
Geometrically, it represents the area under the curve of f(x) between x = a and x = b.
The Riemann Definition
The definite integral is formally defined using Riemann sums. The interval [a, b] is divided into small subintervals, and the sum of the areas of thin rectangles approximates the total area:
∫ab f(x) dx = lim Σ f(xᵢ*) Δx
If this limit exists, the function is said to be integrable on [a, b].
The Fundamental Theorem of Calculus
This theorem connects differentiation and integration. If F(x) is an antiderivative of f(x), meaning F′(x) = f(x), then:
∫ab f(x) dx = F(b) − F(a)
This theorem makes evaluating definite integrals straightforward.
Methods of Evaluating Definite Integrals
1. Using Antiderivatives
The most common method: find an antiderivative and apply the Fundamental Theorem of Calculus.
2. Numerical Integration
For functions without closed-form antiderivatives, numerical methods are used:
- Trapezoidal Rule
- Simpson’s Rule
- Midpoint and Rectangle Methods
3. Substitution
A change of variables can simplify the integral.
4. Integration by Parts
Useful for products of functions.
Applications of Definite Integrals
Definite integrals appear in many fields:
- Physics: work, energy, charge, flux
- Engineering: signal processing, fluid flow, structural analysis
- Economics: total cost, total revenue, consumer surplus
- Mathematics: area, volume, arc length
Example
Compute the area under f(x) = x² from 0 to 2:
∫02 x² dx = [x³ / 3]02 = 8/3
Important Notes
- A function does not need to be continuous everywhere to be integrable.
- If f(x) ≥ 0, the definite integral equals the geometric area.
- If f(x) is negative, the integral gives a signed (algebraic) area.
Conclusion
The definite integral is a cornerstone of calculus, allowing us to compute accumulated quantities across intervals. With the Fundamental Theorem of Calculus, numerical methods, and integration techniques, definite integrals become powerful tools for solving real-world and theoretical problems.