Integration - Indefinite and Definite Integrals

What is an Integration?

Integration is the inverse process of differentiation. Instead of differentiating a function, we are given the derivative of a function and asked to find its primitive, i.e., the original function. Such a process is called integration or anti-differentiation. Suppose we differentiate the function y = x². We obtain dy/dx = 2x. Integration reverses this process, and we say that the integral of 2x is x². 

The situation is more complicated because there are lots of functions, we can differentiate to give 2x. Here are some of them: x² + 4, x² − 15, x² + 0.5. All these functions have the same derivative, 2x, because when we differentiate the constant term, we obtain zero. Consequently, when we reverse the process, we have no idea what the original constant term might have been. So, we include in our answer an unknown constant, c, called the constant of integration. When we want to integrate a function, we use a special notation: ∫ f(x) dx. The symbol for integration, ʃ, is known as an integral sign.

To integrate 2x, we write ʃ 2x dx = x2 + c. Note that along with the integral sign, ʃ, there is a term of the form dx, which must always be written, and which indicates the variable involved, in this case x. We say that 2x is being integrated with respect to x. The function, 2x being integrated, is called the integrand. Technically, integrals of this sort are called indefinite integrals, to distinguish them from definite integrals. When finding an indefinite integral, the answer should always contain a constant of integration.

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: