My Calculus book says:
"We know that the area under a curve $y=F(x)$ from $a$ to $b$ is $A=\int_a^b F(x) dx$, where $F(x)\geqslant0$. If the curve is traced out once by parametric equations $x=f(t)$ and $y=g(t)$, $\alpha \leqslant t \leqslant \beta$, then we can calculate an area formula by using the Substitution Rule for Definite Integrals as follows:
$A=\int_a^b y dx =\int_\alpha^\beta g(t)f'(t)dt\,\,\,\,\,\,\,\,\,\,$[or $\int_\beta^\alpha g(t)f'(t)dt$] ".
Also, it never says that if we have $dx/dt=f'(t)$ we can split the ratio of differentials and obtain $dx=f'(t)dt$.
Could someone deduce the formula of the area under a parametric curve without the use of this mysterious rule of "splitting" the ratio of differentials?
The book deduced the Substituition Rule without doing the split, but said that we could use the split as some non-proved equation to remember how to use this rule. But it is not helpful to me to understand how Substituition Rule was used to obtain the area under parametric curve without using this "split trick".