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					WHY (SINGLE) INTEGRATION IS NEEDED WHEN THERE IS
ALREADY MULTIPLICATION TO CALCULATE AREA?




Area of the figure above = (x2 – x1) × (y2 – y1)
In the above figure, the height of the figure was same i.e. (y2 – y1)
through out the length of the figure i.e. (x2 – x1). So, multiplication was enough to address the situation.




Now, we have to find the area enclosed by the curve with the x-axis and this time,the height of the figure
or, ‘y’ ordinate is not same throughout the length as in previous case.
So, to get over the problem of the height not being constant,
what we do is we select the rectangle (called strip in figure) so thin in length that its height is constant,
then only multiplication is possible as y × dx or as according to convention written as y dx.




Now, consider the slight modification to just earlier figure where the thin strip is taken in the middle .
We can see that this thin strip has also same number of variables i.e. y and dx with exact nature.
You may argue that earlier thin strip has lesser ‘y’ as height and now thin strip has greater ‘y’ (that’s why
they are called variables) but ‘y’ is representing the height of the strip in both cases. So, we call either of
the two figures above as ARBRITARY case and variables there on as ARBRITARY variables.
Now, we can vision the area under the curve figure is formed by the summation
(more appropriately called and should be called integration as summation means summing up the
variables standing at discrete places like height of the figure at x=0, x=1 … and integrating means
summing up the variables standing at continuous places like in the figure above because the figure has
height not only at x=0 and at x=1, but also at all the intermediate points e.g x= 0.1, 0.11,…o.2 as much as it
could exist between x=0 and x=1 )
So, integration should be used here as
Area enclosed by the curve with x-axis = integration of all thin strips
= ∫y dx
since integration is to be done from x1 to x2 , x1 and x2 are placed in upper and lower sides of ∫
respectively ie. integration is done from upper limit x=x2 to x= x1.

				
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Description: Why integration is needed.