What is an integral?
In mathematics, we talk about two types of integrals. Definite integrals and indefinite integrals.
Generally, an integral assigns numbers to functions in a way that can describe displacement, area, volume and even probability.
This type of integral relates to numerical values. It is used in pure mathematics, applied mathematics, statistics, science and many more. However, the very basic concept of a definite integral describes areas.
The definite integral of a function
The symbol used to represent this area
#diamond f " is called the integrand"#
#diamond a and b " are the lower and upper bounds"#
#diamond x " is a dummy variable"#
You might be wondering what
When we say the area defined by the function
If the graph of the function is above the x-axis, then it is said that the net area is positive. If it is below, the net area is negative. This might be harder to grasp at first. This is visualised below:
For example, say we are tasked with finding the net area under the curve
In our case,
To not complicate this answer, here's a video describing it in greater detail:
As such, it has been proven that
We can make a general case here; for every
At the same time, the video describes Riemann sums. These are used to compute integrals. Generally, the Riemann sum of a function
If we remember the general case formed earlier, about the integral of
These are represented as integrals with bounds. Let
You can think of indefinite integrals as generalisations of definite ones.
Instead of being defined by areas, volumes or something else, indefinite integrals correlate to derivatives. The indefinite integral of a function
The Fundamental Theorem of Calculus bridges the gap between a function, its derivative and its indefinite integral. Basically, it says that
Now, say we want to find the antiderivative of the function
Using our former definition, what function do we have to differentiate to get
Except that this is not complete. Remember that, when differentiating a constant with respect to a variable, it practically dissapears, hence the true form of
Analogously, if we define
Bridging the gap between definite and indefinite integrals
Our previous antiderivative of
But we know that this is also equal to
This is where the connection between definite integrals and indefinite integrals is visible, stated formally below:
I hope this answer wasn't too intimidating.