1. WHY Integrals work...

The thing is, I understand that Anti-Derivatives equal net area. I understand that Derivatives equal slope, and I understand how Summations where n is equal to infinity are equal to area. The thing is, I understand WHY Derivatives are slope and why Summations estimate area ... I don't know WHY the Anti-Derivative is equal to net area. It seems like someone just discovered that Anti-Derivatives equal area and so they ran with it .

e.g. Derivatives are technically variations on the slope equation where the distance between points is narrowed down to zero (by limits). Summations estimate area, and so an infinite number if rectangles (n) equals an infinitely accurate estimation (the true area). I don't get how taking the Anti-Derivative is area though, anyone know? Thanks dudes, I appreciate the help .

2. Originally Posted by Thikr
The thing is, I understand that Anti-Derivatives equal net area. I understand that Derivatives equal slope, and I understand how Summations where n is equal to infinity are equal to area. The thing is, I understand WHY Derivatives are slope and why Summations estimate area ... I don't know WHY the Anti-Derivative is equal to net area. It seems like someone just discovered that Anti-Derivatives equal area and so they ran with it .

e.g. Derivatives are technically variations on the slope equation where the distance between points is narrowed down to zero (by limits). Summations estimate area, and so an infinite number if rectangles (n) equals an infinitely accurate estimation (the true area). I don't get how taking the Anti-Derivative is area though, anyone know? Thanks dudes, I appreciate the help .
From the attachment

$f(x)=\sum\Delta{y}$

$As\;\;\;\Delta{x}\rightarrow\ 0,\;\;\;f(x)\rightarrow\ \int{}dy$

From differentiation...

$\displaystyle\frac{dy}{dx}=\frac{d}{dx}f(x)=f'(x)\ \Rightarrow\ dy=dx\frac{d}{dx}f(x)=f'(x)dx$

$\displaystyle\int{}dy=y=\int{\frac{d}{dx}f(x)}dx=\ int{f'(x)}dx$

3. You might also like to look at Wikipedia's article on the Fundamental Theorem. There are some really great geometric explanations.

Fundamental theorem of calculus - Wikipedia, the free encyclopedia