Indefinite and Definite Integrals

What is the Difference between Indefinite and Definite Integrals?

Indefinite Integrals: If the function F(x) is an anti-derivative of f(x), then the expression F(x) + C, where C is an arbitrary constant, is called the indefinite integral of 𝑓(𝑥) with respect to x and is denoted by ∫ 𝑓(𝑥) 𝑑𝑥, i.e., ∫𝑓(𝑥)𝑑𝑥 = F(x) + C. The function f(x) is called the integrand, ʃ the integral sign, x is called the variable of integration, and C is the constant of integration.

Definite Integrals: A definite integral is simply an indefinite integral, but with numbers written to the upper and lower right of the integral sign. A definite integral is usually a number. We define the indefinite integral of the function 𝑓(𝑥) with respect to 𝑥 from 𝑎 𝑡𝑜 𝑏 to be:

Where, 𝐹(𝑥) is the anti-derivative of 𝑓(𝑥). We call 𝑎 𝑎𝑛𝑑 𝑏 the lower and upper limits of integration, respectively. The function being integrated, 𝑓(𝑥), is called the integrand. Note that integration constants are not written in definite integrals since they can always be cancelled.

A Definite Integral as an Area under a Curve:

The definite integral of a function 𝑓(𝑥) which lies above the 𝑥-axis can be interpreted as the area under the curve of 𝑓(𝑥). Thus, the area shaded in the figure is given by the definite integral:

Consider the area, A, under the curve, y = f(x). The area below the curve is an antiderivative or integral of the function.

Example:

Consider the integral:

The integrand 𝑦=𝑥 (a straight line) is sketched below. The area underneath the line is the shaded triangle. The area of any triangle is half its base times its height. For the shaded triangle, this is

As expected, the integral yields the same result: