Integration is a very important concept in calculus, It is a handy mathematical application that we can employ to calculate areas under any curve or to decipher the volume of a shape.
Intuitively then, an integral value would be described as designating a number in a function such that it describes the area, volume and/or displacement etc.
There are two main kinds of integrations, called definite integration and indefinite integration.
On the other hand, there is an Indefinite integral which lacks endpoints and reaches infinity, it is also sometimes known as improper integral.
Integrals can also be generalized depending on the type of function along with the domain over which the mathematician carries out the integration.
Consider this, a line integral describes a function of two or more than two variables, and the interval related to integration gets replaced by a curve joining the two endpoints in the interval.
When it comes to surface integral however, the curve gets replaced by part of a surface in 3 dimensional space.
The definite integral pertaining to limits is as follows:
represents the integral
a and b are the limits that in this context, represent the endpoints of the bounded region that needs to be measured.
dx represents the differential of the variable ‘x’
The fx at last, represents the integrand.
In order for you to do this, you must first have to decide on the order in which you plan to carry out the integration.
There are examples where this does not really matter a lot, and those examples in which the difference in the ease of carrying out the integral is very meaningful.
However, you can always intend to change the order of integration if you are not really satisfied with your choice.
So consider we have an area of integral and we make sure to integrate over the x variable first and then y.
The fundamental step in deciding on limits of integration is to sketch out a picture of the region you wish to integrate over. This region would usually be bounded/enclosed by a set of curves.
For certain choices of the variable y, the integration limits of x will usually be the values of x that are on two of these bounding curves for our y value.
You then integrate over y for those intervals of its values for which you obtain the same bounding curves in x. You can then fix a y value in each of these intervals.
In most cases that you’d come across, the values of x that you wish to integrate over, will form an interval lying somewhere within two of these curves.
You must then find out which curves these are (sometimes they can be identical) and then solve for each curve equation for its x value with assumed y value. These would turn out to be the limits for your x integral for the y value.
In certain cases, the limits on x would involve varying curves for varying values of y. You must choose your x integral accordingly.
If you draw a picture, it is usually very plainly simple to do. Without a picture then, it would become ridiculously difficult to get it right and you’d more likely mess things up. e