Integral Definition
Integral is defined as how much is something accumulating, gathering, or summing up. Like you download a file from the internet it means you are accumulating data from the internet to your PC.
Similarly, download the second movie in the second hour and the third in the third hour. Now I ask you that how much data you gather on your PC in three hours is the main question regarding the integral.
Let’s create a graph between speed and time to accumulate data on your PC for understanding integral definition graphically.
Here we can see that the data accumulated in between the speed and time graph is an enclosed area, it shows the integral always tells about data of any area enclosed.
Most of the rare cases we think have areas enclosed in a straight or rectangular form like a car moving with constant speed or downloading data with constant internet speed which we can calculate easily.
But, in most cases, the straight area does not happen.
Area Between Curves
Can you think you are traveling in a car and get back home after an hour with the same and constant speed during the whole time? Definitely Not.
Everything will never be constant, it will be changed. Similarly, traveling speed may be increased or decreased with time as given in the graph.
In more general for two variables y and x which are related to each other in some way like y= f(x). By changing xthere will be change occur in y, but now the area enclosed will never be a straight area. It is an area between curves, and we don’t know how to calculate the area under curves.
After a long interval of research Isaac Newton and Leibniz found the concept for finding the area under the curve which further led to the formation of the integral definition. This definition leads to the concept for area between curves.
Concept of integral
Before the concept of integral, people used the method of dividing the area under the curve into small rectangles or squares and calculating the areas of rectangles and accumulating all these.
Using the same method if we break the area under the curve of the function into two pieces then the area will be
$$ \text{Total area} \;=\; \text{Area of 1st rectangle} \;+\; \text{Area of 2nd rectangle} \;+\; \text{Shaded area} $$
Now we can calculate the area of rectangles at a given function in such a way
$$ \text{Total area} \;=\; f(x_1)Δx + f(x_2)Δx + \text{Shaded area} $$
Similarly, if we divide the area into an infinite number of rectangles in such a way that shaded area≃0 which can be neglected. So area under the curve will be,
$$ Total Area = \sum_{i=1}^N f(x_i)Δx $$
If these chunks of rectangles have such a small width so that x≈0
$$ \text{Total area} = \lim\limits_{x \to 0} \sum_{i=1}^N f(x_i)Δx $$
But this idea looks great in paperwork only. It will be extremely painful if someone has to add the area of infinite rectangles manually. But fortunately, we found a shortcut in the 17th century, and that shortcut became a concept of integral.
And now we basically see what was that shortcut that introduces the concept of integration in Mathematics.
Integration in Mathematics
Let’s suppose we have a function y=f(x), as shown in figure.
So, if we increase x by Δx then the area will also be changed as A(x) by ΔA.
Now for a small change, if we know a relation in function then we can calculate its derivative. If we calculate derivative in our situation then
$$ \text{Derivative of Area} = \frac{d(A)}{dx} $$ $$ \implies \lim\limits_{x \to 0} \frac{ΔA}{Δx} = \frac{f(x)Δx}{Δx} $$
So, what we found that derivative of the curved area is that function itself,
$$ A'(x) = f(x) $$
So if we have a function f(x) whose area will be calculated, we have to only find a function F(x) whose derivative is the same as that function f(x), then F(x) is the area of that function.
In general, if we find any sub-function F(x) of our given function f(x) whose derivative is our given function f(x), then this function f(x) is called integration in mathematics and the function F(x)will be called an integral.
So now we have to know how to find F(x) ?
Many mathematicians find that F(x) for many instances for us, that we already learned in our calculus classes as a formula like,
$$ \int x^n dx = \frac{x^n+1}{n+1} + c \;\;\;\;\;\;\;\; where \;\; n \gt 0 $$ $$ \int sin(x) \;=\; -cos(x)+c $$
where c = constant of integration
Similarly there are many other standard integral functions that are calculated by scientists to make integration easy and reusable for us.
It is added because many functions contain the same derivative which have constants in it and are neglected by using the rule of derivatives.
Integral Notation
The integral symbol is just look like as S in stylish format "" which evolved from S in sum, which came from the summation means gathering or summing up anything. Such type of S symbol is used for integral notation.
E.g.
$$ \int sin(x) dx $$
Here, ∫ is the integral symbol which is used for representing integral notation.
Next we put sinx which is a function we want to integrate, that function is called Integrand.
At the end we put dx in the notation which denotes that the integrand is approaching zero in the direction of “x”.