Monday, 18 August 2014

INTEGRATION EXPLAINED INTUTIVELY



Okay...there is a slight mistake. It should be volume of the cylinder and not surface area of the cylinder.

TAKING ANOTHER EXAMPLE.

GOING FURTHER AND BUILDING A BRICKWALL, BUT THIS TIME WE WILL REQUIRE DOUBLE INTEGRATION. WHY?










SIMILARLY, TRIPLE INTEGRATION CAN BE USED TO BUILD A FORT OR A ROOM WITH THE HELP OF ANOTHER DIMENSION dz.

MORE IN DETAIL

On the 25 December 1642, Christmas Eve at 12.00pm…….Isaac Newton was born. 

Unlike Normal children who are born in 9 months, young Isaac was born in just 6 months (Such children are called as pre-matured children). The doctor told his mother that her child had barely survived. 

 He added that such children have a stunted physical growth and will have severe problems in learning and writing as they grow up.
It was true initially. 

Young Isaac Newton had problems grasping answers in his early school life. However later he was not only to grasp them but form his own generation of Physics and Mathematics that forms the basis of Mathematics and Physics even till today.


Newtonian Physics( Classical Physics) ruled for about more than 250 years until Albert Einstein formed Modern Physics.

However, his contribution towards Mathematics in the form of Integration (Calculus) is an underrated contribution given by him. It simplified a lot of things for mathematicians.

DID YOU KNOW???? 







              
Legend has it that while Isaac Newton had just written half the book....a dog of his put the book on fire in the chimney. However in just a period of 6-months he re-wrote the whole book!!!!. 


 Now what is integration after all?
 In short Integration is a short-cut to multiplication.
 Now multiplication itself being a short-cut to addition, one can say that Integration is a short cut to addition (summation).

In order to understand integration intuitively.....just keep asking that if you keep adding something in a certain direction.....what will you get?

The direction in which you keep adding is given by dx (along the x-axis direction) , 
dy (along the y-axis direction),
dh( along the height),
d0 ( along a curve of angle 0 ).

Between the integration sign and the dx (or dy or dh or d0),....whatever you put is the magnitude of thing ( I visualize it as a block)  you are putting.

So when you do integration of s2dx...."2" is the magnitude.
"dx" means direction x (along x-axis).
Answer = 2x.

Interpretation of Answer..... 
Lets say, you kept adding blocks of 2kilos.....5 times. So x=5.
Answer = 2.x...means 2 x 5 =10.
So if you add blocks of 2 kilos each 5 times....answer is 2 x 5 = 10kilos.

If you add blocks of 2 kilos each 19 times....x becomes 19....so anwer = 2x = 2x19=38.

'x'.....defines how many times or to what extent you kept adding.



RELATED IMAGES
  
LINK HERE










































AREAL INTEGRAL LINK HERE














GRAPHICAL INTERPRETATION OF INTEGRATION













MATH STACK EXCHANGE LINK HERE


MATHS IS FUN.COM.....LINK HERE ON INTEGRATION.



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ABOUT

I hardly understood Math in school. In fact was on the verge of dropping the subject I loved the most because as much as I loved the theory of it, I could not understand the math involved in it. However I loved the subject too much to be able to live without. Hopelessly, I was continuing my love-affair with it. 


Then one day....a miracle happened,....while applying a certain formula again and again.....I came to know its significance. Slowly and steadily....other equations also started clicking. I got to see a strong relationship between Maths and the Physics it was pointing towards. 

They both were the same. Maths was just an easy language to express a physical phenomenon. And a bit more to that in the sense that it could even predict the behaviour of a certain physical phenomenon. Equations now as if came to life.  Every equation now had as if something to say. A burning urge to share these things with the world aflamed within me. So that no one has to give up the subject that he or she loves the most. 


The book on visualizing maths thus got written as a sprout of inspiration. The blog followed. Both these are dedicated to you and all such similar minds searching for answers.Visualizing Math & Physics is a blog dedicated to you and all such similar minds searching for answers
.

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VISUALIZING MATHS.PDF




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