fundamental theory of calculus

[Tim_Lou]

this is just a thread to see how many people we will have if we open a new calculus group.

has anyone ever tried to prove the integral of a function is the anti-derivative of it?
i did it using mean value theorm.... did you prove the theorem using someway else?

[Bo]

a very fundamental proof is something like this:

F: indefinite integral of f
G(x)=int(a->x){f(t)dt) : indefinite integral of f

F(x)=A(x)+C (always exist, C=constant)
calculate F(a)-F(b), and realize that G(a)=0.

Bo

[sanctus]

What is the anti-derivative?

[Bo]

the antiderivative is the same as the indefinete integral... just 2 words for the same thing.

Bo

[Tim_Lou]

Originally Posted by Bo

a very fundamental proof is something like this:

F: indefinite integral of f
G(x)=int(a->x){f(t)dt) : indefinite integral of f

F(x)=A(x)+C (always exist, C=constant)
calculate F(a)-F(b), and realize that G(a)=0.

Bo

Bo, how is it able to prove that the "area" under a curve is the antiderivative of that curve?
maybe i asked the wrong question?

[Tim_Lou]

what i did is basically...
mean of f'(x) = (f(b) - f(a)) / (b-a)
"area" = mean height * base
so..... "area" = (f(b) - f(a)) / (b-a) * (b-a) which is integral of f'(x) from a to b

*"area" isnt really area, its actually the sum of it...
the prove is simple, and... not very supportive