The Fundamental Theorem of Calculus

[Chump]

Homework Statement

I'm having trouble wrapping my head around this concept. I understand integration and differentiation individually. I even understand the algebraic manipulations that reveals their close relationship. However, the typical geometric interpretation of a 1-D curve being the derivative of the area function below it seems odd to me. I'm trying to get an intuitive understanding of how a 1-D curve, say A'(x) = f(x) , is the derivative of an area function, A(x). I think I might know where my source of confusion lays: When dealing with derivatives, I'm used to visualizing in terms of tangent lines drawn to a 1-D curve. It seems weird to apply it to an area function, which is more irregular polygon than it is 1-D curve. I hope my concerns make sense.

Homework Equations

The Attempt at a Solution

 

[BvU]

Googling doesn't help ?

 

[William White]

try googling

mit Session 47: Introduction of the Fundamental Theorem of Calculus

and watch that

 

[ChristianQG]

+ S[x+dx] = S[x] + f(x)dx. Obvious from geometry. => f(x)=S'(x)

 

[Chump]

try googling

mit Session 47: Introduction of the Fundamental Theorem of Calculus

and watch that

Great video, but it didn't quite answer my question. I might not have been able to articulate my question that well over the internet. Oh, well. Thanks, anyhow.

 

[geoffrey159]

I think the missing link between integration and differentiation is primitiv(-isation,-ization, -ation, ..., I don't know). It is the 'reciprocal' operation to differentiation: $\phi$ is a primitive of a piecewise continuous fonction $f$ if and only if $\phi$ is differentiable on the domain of $f$ and $\phi' = f$.
The set of all primitives of $f$ vary by an additive constant. So all primitives of $f$ have the form $\phi = \phi_0 + C$, where $\phi_0$ is a particular primitive of $f$.
It happens that the function $\phi_0(x) :=\int_{x_0}^x f(t) dt$ is a primitive of $f$ wherever $f$ is continuous (just show that if $f$ is continuous in $a$, $\frac{\phi_0(x) - \phi_0(a)}{x-a} \rightarrow f(a)$ as $x\rightarrow a$ ). I think it explains a little bit the link between integration and differentiation.The link between the area under the curve of $f$, defined on $[a,b]$, and its integral, comes from the construction of the integral of piecewise continuous functions from the integral of step-functions on $[a,b]$, which are homogenous to an area