PlanetPhysics/Fundamental Theorem of Integral Calculus

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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:


Converse[edit]

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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