Definition of Integration
Integration in calculus is the mathematical process of finding the integral of a function, which represents the antiderivative or the accumulation of quantities. It is the inverse operation of differentiation, where the derivative measures instantaneous rates of change, and integration computes total change over an interval. Indefinite integrals yield a family of functions differing by a constant, while definite integrals provide a numerical value representing net accumulation.
Key Principles of Integration
The fundamental theorem of calculus links integration and differentiation, stating that the definite integral of a function's derivative over an interval equals the function's value at the endpoints. Basic rules include the power rule for integrating polynomials, where ∫x^n dx = (x^{n+1})/(n+1) + C for n ≠ -1, and linearity, allowing integration of sums and scalar multiples. Substitution and integration by parts handle more complex forms.
Practical Example: Integrating a Quadratic Function
Consider the function f(x) = 2x + 1. The indefinite integral is ∫(2x + 1) dx = x^2 + x + C, where C is the constant of integration. For a definite integral from 0 to 3, evaluate [x^2 + x] from 0 to 3: (9 + 3) - (0 + 0) = 12. This represents the area under the line y = 2x + 1 from x = 0 to x = 3.
Importance and Applications of Integration
Integration is crucial for solving real-world problems, such as calculating areas and volumes in geometry, determining displacement from velocity in physics, and modeling population growth or economic quantities in applied sciences. It underpins advanced topics like differential equations and probability distributions, enabling precise quantification of continuous changes in engineering, biology, and finance.