Integral Calculus is the branch of calculus where we study about integrals and their properties. Integration is a very important concept which is the inverse process of differentiation. Both the integral calculus and the differential calculus are related to each other by the fundamental theorem of calculus.
If we know the f’ of a function which is differentiable in its domain, we can then calculate f. In differential calculus, we used to call f’, the derivative of the function f. Here, in integral calculus, we call f as the anti-derivative or primitive of the function f’. And the process of finding the anti-derivatives is known as anti-differentiation or integration. As the name suggests, it is the inverse of finding differentiation. Integration can be classified into two different categories, namely,
An integral that contains the upper and lower limits (i.e) start and end value, then it is known as a definite integral. On a real line, x is restricted to lie. Definite Integral is also called a Riemann Integral when it is restricted to lie on the real line.
A definite Integral is represented as:
Indefinite integrals are not defined using the upper and lower limits. The indefinite integrals represent the family of the given function whose derivatives are f. It returns a function of the independent variable.
The integration of a function f(x) is given by F(x) and it is represented by:
∫f(x) dx = F(x) + C
where R.H.S. of the equation means integral off(x) with respect to x
F(x) is called anti-derivative or primitive.
f(x) is called the integrand.
dx is called the integrating agent.
C is called constant of integration.
x is the variable of integration.
It may seem strange that there exist an infinite number of anti-derivatives for a function f. Taking an example will clarify it. Let us take f’ (x) = 3x2. By hit and trial, we can find out that its anti-derivative is F(x) = x3. This is because if you differentiate F with respect to x, you will get 3x2. There is only one function that we got as the anti-derivative of f. Let us now differentiate G(x)= x3+9 with respect to x. Again we would get the same derivative i.e. f.
This gives us an important insight. Since the differentiation of all the constants is zero, we can write any constant with x3 and the derivative would still be equal to f. So, there are infinite constants which can be substituted for c in the equation F(x) = x3+ C. And hence, there are infinite functions whose derivative is equal to f. And hence, there are infinite functions whose derivative is equal to 3x2. C is called an arbitrary constant. It is sometimes also referred to as the constant of integration.
Integral Calculus is mainly used for the following two purposes:
1. To calculate f from f’. If a function f is differentiable in the interval of consideration, then f’ is defined in that interval. We have already seen in differential calculus how to calculate derivatives of a function. We can “undo” that with the help of integral calculus.
2. To calculate the area under a curve.
Until now, we have learned that areas are always positive. But as a matter of fact, there is something called a signed area.
Just like we had differentiation formulas, we have integral formulas as well. Let us go ahead and look at some of the integral calculus formulas.
The important application of integral calculus are as follows. Integration is applied to find:
Below are the examples of integration Calculus: