User:Chili b/Fundamental Theorem of Calculus
This isn't a proof (you can find that elsewhere). It's just some writing to help myself with my intuitive understanding.
Given a function [math]\displaystyle{ f }[/math] whose domain is all the real numbers, the definite integral [math]\displaystyle{ \int^b_a f(x) ~ dx }[/math] (for [math]\displaystyle{ a, b \in \mathbb{R} }[/math]) can be understood geometrically as the area under the graph of [math]\displaystyle{ f }[/math] between [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math].
We could approximate this area by packing a bunch of rectangles under the graph of [math]\displaystyle{ f }[/math] such that each rectangle's height is equal to the value of [math]\displaystyle{ f }[/math] at that rectangle's left edge. This leads to the idea behind the Riemann sum which is that if we reduce the sizes of the rectangles, the approximation gets more accurate so if we make the rectangles infinitely small, then the sum of their areas will accurately describe the area under the graph of [math]\displaystyle{ f }[/math].
Therefore, we can express the integral with this limit: [math]\displaystyle{ \lim_{\delta \to 0^+} \sum^n_{i=0} f(a + \delta i) \cdot \text{min}(b,~ a + \delta(i + 1))~ }[/math] where [math]\displaystyle{ n = \left\lceil \frac{b - a}{\delta} \right\rceil }[/math], i.e. take the sum of the areas of rectangles packed under the graph of [math]\displaystyle{ f }[/math] which have at most width equal to some infinitesimally small [math]\displaystyle{ \delta }[/math].
The Fundamental Theorem of Calculus (FTC) says [math]\displaystyle{ \int^b_a f(x)~ dx = F(b) - F(a) }[/math] where [math]\displaystyle{ F }[/math] is an antiderivative of [math]\displaystyle{ f }[/math], i.e. [math]\displaystyle{ f }[/math] is the derivative of [math]\displaystyle{ F }[/math]. When we differentiate, we lose some information as constants get dropped (the derivative of a constant is 0), but this doesn't matter in this case because we are taking the difference of [math]\displaystyle{ F(b) }[/math] and [math]\displaystyle{ F(a) }[/math] so the constant parts will cancel out (they are equal, so their difference will evaluate to 0).
Let's go back to the idea of the Riemann sum and combine it with the idea that [math]\displaystyle{ f }[/math] represents the rate of change of some function [math]\displaystyle{ F }[/math]. If the height of each rectangle is the rate of change of the function [math]\displaystyle{ F }[/math] at some point, then the area of that rectangle is how much that [math]\displaystyle{ F }[/math] would change if its rate of change remained constant for the width of the rectangle. Therefore, the areas of all the rectangles approximate how much [math]\displaystyle{ F }[/math] changes between [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math] (the smaller the rectangles, the closer to the actual change it is). Therefore the Riemann sum and consequently the integral represents exactly how much [math]\displaystyle{ F }[/math] changes from [math]\displaystyle{ a }[/math] to [math]\displaystyle{ b }[/math], which is what is expressed by the FTC.