PlanetPhysics/Fundamental Theorem of Integral Calculus

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:

ConverseEdit

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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