Consider the sequence of numbers
and define the difference
. Now sum the differences and not that all but the first and last terms cancel:
In other words
. It seems obvious that,
Changing variables:
or as an indefinite integral:
In other words, the integral of the derivative of a function is the original function. But what of the derivative of the integral? Let,
where .
Here, we assume that all the intervals in the Riemann sum are equal. To find we need to add one extra term to the Riemann sum:
- .
As shown in red, the change in area (∫fdx) of a function is closely related to the value of the function, f(x) at the point where x changes to x+δx
.
Rearrange this to obtain:
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