Calculus - The definite integral and its applications — Definite integration

About this lesson

This unit introduces the concept of the definite integral, building on the indefinite integral. It explains that while the indefinite integral includes an arbitrary constant, the definite integral, evaluated from a lower limit a to an upper limit b, yields a definite real number. The fundamental theorem of calculus is presented as the key tool: if f is continuous on [a, b] and F is any antiderivative, then the definite integral equals F(b) - F(a). The unit covers notation, properties (such as reversing limits, zero-width intervals, and additivity over adjacent intervals), and techniques for evaluation, including direct application of standard integrals and substitution. It also addresses integrals involving absolute values and discusses special properties for odd and even functions over symmetric intervals, stating that the integral of an odd function over [-a, a] is zero and that of an even function is twice the integral from 0 to a. Worked examples and exercises involve polynomial, trigonometric, exponential, and rational functions, with some problems requiring the use of modulus functions and substitution.