First Fundamental Theorem of Calculus question regarding F'(x) The first fundamental theorem is defined:
$F(x) = \int^x_c f(t) dt$ where $c$ is constant.
The corollary:
$\int^b_a f(t) dt = F(b) - F(a)$
My question is how can the integral $[c, x]$ in the first theorem be calculated without the corollary?  Otherwise would $c$ be assumed to be $0$?
As an example:
$f(t) = t^2$,
$F(t) = \int^x_c f(t) dt = \int^x_c t^2 dt$
Antiderivative:
$F(t) = \frac {t^3} 3$
However, my understanding is the antiderivative is the area under the curve from $t = 0$.
Appreciate any guidance.

You are skipping a step. The corollary can be stated:
For a function $F'(x)=f(x)$ (such a function is called an antiderivative)
$$\int_{a}^{b}f(t)dt=F(b)-F(a)$$
Hence it tells you that in order to calculate the definite integral you need to find an antiderivative $F$.
