In the last post, we studied integration and some basic rules for finding integrals. In this post, we try to learn a lot more about the concept of integration.
There are two types of Integrals –
A] Indefinite Integrals.
B] Definite Integrals.
Indefinite Integrals
While specifying the indefinite integral, no upper and lower limits are defined. After integrating the function under study, we get an answer that still has the ‘x’ term/s (i.e.variables) and the constant of integration ‘C’. So, the answer is NOT a definite number.
It can be represented as –
e.g. ∫ x3.dx = ?
We know the general rule from last post that ,
∫xn = [(xn+1)/n+1] + C ⇒ (These are general mathematical rules and we just follow them).
∴∫x3.dx = [x3+1 / 3+1] + C
= [x4 / 4] + C ⇒ This answer has the ‘x4′ term in it and thus it’s not an absolute number.
So, the above integral is an example of an indefinite integral.
Definite Integrals
In this case, the upper and lower limits on the integral are specified. Definite Integrals are generally represented as follows –
As the upper limit (b) and the lower limit (a) is specified, definite integrals yield a specific number as the answer.
e.g.–
In the above example, x can take values only in the range from 1 to 2. The range can be found out by subtracting the higher limit (i.e plugging in x=2) from the lower limit of the answer (found by plugging in x=1) as follows –
Solving this integral gives us a definite answer – A NUMBER! Thus, this is a definite integral. Here the constant of integration is nullified. Thus we have NO CONSTANT OF INTEGRATION.
In a geometric sense, integration can be interpreted as the ‘area under the curve‘. As we add up the infinitesimal small changes in a quantity, over a period of time, we get the area under the curve. This area represents the summation of many small areas on the graph. The area under a curve between two points can only be found by using a definite integral between the two points. This is because area has to be a number – a finite value and only definite integrals give us a number as the answer.
e.g. Find the area bounded by the lines y = 0, y = 1 and y =x2
In the above example, the function is y =x2. This means that y exponentially increases with x. Thus, the curve above is exponential. Integrating this function and putting the limits y=0 and y=1, we can find out what area this function occupies in the curve.
To find the area we first integrate the function y= x2 -4. Then, we plug in x=2 and x= -2 and subtract the upper limit from the lower limit to find the range, i.e. area under the curve.
I conclude my posts on calculus with this post. I sincerely hope that these basic concepts will help us understand many topics in physical chemistry in a better way.
Be a perpetual student of life and keep learning …
Good day !
Madoverchemistry 