In the last post we studied what Integration means and some basic rules for finding integrals.I mentioned a term *‘Indefinite Integral’* in my last post.In this post let us discuss this term and other concepts related to Integration.

There are two types of Integrals –

**A]Indefinite Integral. **

**B]Definite Integral.**

**A]Indefinite Integral – **While specifying 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) in it and the constant of integration ‘C’. So, the answer is **NOT** a definite number.

It can be represented as –

**∫f(x).dx**

*e.g.* ∫x^{3}.dx = ?

We know the general rule from last post that ,

**∫x ^{n} =**

**[(x**

^{n+1}**)/n+1] + C**⇒ (These are general rules and we just blindly follow them).

∴∫x^{3}.dx = [x^{3+1} / 3+1] + C

= [x^{4 } / 4] + C ⇒ This answer has ‘x^{4′ }term in it so it’s not an absolute number.

So, the above integral is an example of Indefinite integral.

**B]Definite Integral – **In this case,we specify the upper and lower limits on the integral.So,we get a definite number as the answer.

It can be represented as –

*e.g.*

Here we have specified the limits for our answer. This means that x can take values only in the range from 1 to 2. So how much is this range ? We can easily find out by subtracting the higher limit of the answer (i.e plugging in x=2) from lower limit of our answer(i.e plugging in x=1) as follows –

Now, we have got a definite answer – A NUMBER !Thus, this is a definite integral.Here the constant of integration is nullified and we have NO CONSTANT OF INTEGRATION.

In geometric sense, integration can be interpreted as ‘** area under the curve**‘.As we add up the infinitesimal small changes in a quantity over a time period, we get area , which represents the summation of many small areas on the graph.The area under a curve between two points can be found by doing a

**definite**integral between the two points.Why definite integral ? That is because area has to be a number – a finite value and only definite integral gives us a number as the answer.

e.g.Find the area bounded by the lines y = 0, y = 1 and y =x^{2}

So here the function is y =x^{2 .}Integrating this function and putting limits y=0 and y=1 , we can find out what area this function occupies in the curve which defines this function i.e y=x^{2 .}

To find area we first integrate the function y= x^{2 }-4.Then, we just plugged in x=2 and x= -2 and subtracted the upper limit from 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 shall help us understand many topics in physical chemistry in a better way.

Be a perpetual student of life and keep learning …

Good day !

References and Further Reading –

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