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assid wrote:-
> could someone enlighten us as to the precise relation****p between
> differentiation and integration. what i had in mind when visualising
> the concepts of diff and int was that diff is basically the rate of
> change of one variable with respect to another at a point in the path
> of its function and that int is basically the reverse process whereby
> one obtains the function of the relation****p between two variables in
> the first order from the function at a point in its path.
The common concept uniteing differentiation and integration
is 'measureing via dissapearingly smaller intervals'.
When you continually halve an interval, of say time or length
or any other type, it's size approaches zero. With sufficient halvings
it gets as close to zero as you can specify.
Since as you've shown that you know, differentiation requires
the 2 variables to be considered over some interval.
The derivative 'at a point' means that the interval has become
infinitesimally small. But that's no problem, since as the one
variable is continually halves to approach zero, so does the 2nd
variable change too. Therefore you keep the 'ratio of the 2
variables', which is defined as the derivative.
For integration [with 2 variables] a similar trick is used:
since irregularly shaped areas can't be calculated, the
area is approximated to any required degree of accuracy, by
dividing it into increasingly more, rectangles of decreasing
size/s.
Similar to the case of differentiation:
as the inerval/rectangle-width decreases towards zero,
the correct answer is approached. So that when the
interval approaches zero [size], the error in the answer
approaches zero too.
With diferentiation the 2nd variable moves towards zero
as the 1st varailble does. With integration, the width
moves towards zero as the number of rectangles moves
towards infinity. In both cases, the initial estimate
improves towards perfect accuracy as the intervals
approach zero. Ie. the estimation error approaches zero
== Chris Glur.
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