The original methods to find the area are division, accumulation and limit.
Integral is only used to find the original function of a function.
But later Newton found that finding the area can be linked with finding the integral, and deduced the formula. After that, finding the area became a definite integral and finding the original function became an indefinite integral.
As for the explanation of this formula, it is best to look at advanced mathematics or mathematical analysis. You need to learn some preparatory knowledge, such as riemann sum, variable upper bound definite integral, and then look at the proof of the formula.