I was trying to understand, how does the proof of the FTC1 ties to FTC2. Can someone please correct me, if my reasoning is wrong?
The first part tells us, that if function $f$ is continous over the interval $[a,b]$, then:
[MATH]f(x) = F'(x)[/MATH], where [MATH]F(x) = \int_a^xf(t)dt[/MATH], for every [MATH]x \in [a,b][/MATH]
Now would it be correct (?) to write this as:
[MATH]F'(x) = f(x)[/MATH] // [MATH]\int[/MATH]
antiderivative of both sides, to get rid of the derivative on the left, and we get:
[MATH]F(x) = \int f(x)[/MATH]
And then just plug it in:
[MATH]\int_a^bf(t)dt = \int_a^bf(t)dt - \int_a^af(t)dt = \int f(b) - \int f(a) [/MATH]
So we are left with the notion that a definite integral of function [MATH]f[/MATH] from [MATH]a[/MATH] to [MATH]b[/MATH] is the antiderivative of [MATH]f [/MATH]evaluated at [MATH]b - f[/MATH] evaluated at [MATH]a[/MATH]: which is how we calculate integrals
Is my reasoning good? ( for a beginner of course, this is not meant to be a professional proof )
The first part tells us, that if function $f$ is continous over the interval $[a,b]$, then:
[MATH]f(x) = F'(x)[/MATH], where [MATH]F(x) = \int_a^xf(t)dt[/MATH], for every [MATH]x \in [a,b][/MATH]
Now would it be correct (?) to write this as:
[MATH]F'(x) = f(x)[/MATH] // [MATH]\int[/MATH]
antiderivative of both sides, to get rid of the derivative on the left, and we get:
[MATH]F(x) = \int f(x)[/MATH]
And then just plug it in:
[MATH]\int_a^bf(t)dt = \int_a^bf(t)dt - \int_a^af(t)dt = \int f(b) - \int f(a) [/MATH]
So we are left with the notion that a definite integral of function [MATH]f[/MATH] from [MATH]a[/MATH] to [MATH]b[/MATH] is the antiderivative of [MATH]f [/MATH]evaluated at [MATH]b - f[/MATH] evaluated at [MATH]a[/MATH]: which is how we calculate integrals
Is my reasoning good? ( for a beginner of course, this is not meant to be a professional proof )