Definite Integral as Limit of Sum
The definite integral of f(x) over the interval [a, b], denoted by, is defined as the limit of a sum given by:
where nh = b – a
f(x) is said to be integrable over [a, b] if the above two limits exist and are equal.
Applications of Definite Integrals
Definite Integrals are used to find areas of the complex curve, volumes of irregular shapes, and other things. Definite Integrals are defined by, let us take p(x) to be the antiderivative of a continuous function f(x) defined on [a, b] then, the definite integral of f(x) over [a, b] is denoted by and is equal to [p(b) – p(a)].
= p(b) – p(a)
The numbers a and b are called the limits of integration where a is called the lower limit and b is called the upper limit. The interval [a, b] is called the interval of the integration.
Note
- Constant of Integration is not included in the evaluation of the definite integral.
- is read as “integral of f(x) from a to b”