12.Chemistry & Mathematics VIII-Calculus(5)-Integration(2).

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 IntegralWhile 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. ∫x3.dx = ?

We know the general rule from last post that ,

∫xn = [(xn+1)/n+1] + C ⇒ (These are general rules and we just blindly follow them).

∴∫x3.dx =  [x3+1 / 3+1] + C
= [x / 4] + C ⇒ This answer has ‘x4′ 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 –defint

e.g.Blank White 1_Fotor

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 –

Blank White2 _Fotor

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 =x2volume4

So here the function is y =x2  .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=x2 .

integration1

To find area we first integrate the function  y= x-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 –

1)http://archive.learnhigher.ac.uk/resources/files/Numeracy/Integration_webversion.pdf
2)http://www3.ul.ie/~mlc/support/Loughborough%20website/chap14/14_1.pdf
3)http://www.intmath.com/integration/integration-mini-lecture-indefinite-definite.php
4)http://stat.math.uregina.ca/~kozdron/Teaching/UBC/101Spring98/101integral.html
5)http://revisionmaths.com/advanced-level-maths-revision/pure-maths/calculus/area-under-curve
Images source –
1)http://revisionmaths.com/advanced-level-maths-revision/pure-maths/calculus/area-under-curve

 

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