The first part tells us, that if function $f$ is continous over the interval $[a,b]$, then:

\(\displaystyle f(x) = F'(x)\), where \(\displaystyle F(x) = \int_a^xf(t)dt\), for every \(\displaystyle x \in [a,b]\)

Now would it be correct (?) to write this as:

\(\displaystyle F'(x) = f(x)\) // \(\displaystyle \int\)

antiderivative of both sides, to get rid of the derivative on the left, and we get:

\(\displaystyle F(x) = \int f(x)\)

And then just plug it in:

\(\displaystyle \int_a^bf(t)dt = \int_a^bf(t)dt - \int_a^af(t)dt = \int f(b) - \int f(a) \)

So we are left with the notion that a definite integral of function \(\displaystyle f\) from \(\displaystyle a\) to \(\displaystyle b\) is the antiderivative of \(\displaystyle f \)evaluated at \(\displaystyle b - f\) evaluated at \(\displaystyle a\): which is how we calculate integrals

Is my reasoning good? ( for a beginner of course, this is not meant to be a professional proof )