PlanetPhysics/Fundamental Theorem of Integral Calculus

From Wikiversity
Jump to navigation Jump to search

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 | edit source]

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:

The original Planet Physics version still remains on the page as a hidden comment (visible in "edit" mode)