How exactly are definite integrals calulated and what is the connection between definite and indefinite integrals? [duplicate]

It's taught that an indefinite integral or an antiderivative of a function gives the function that when differentiated returns the function that was integrated, mathematically represented by the following: $$\int f(x)dx = F(x)$$ $$F'(x) = f(x)$$ and that a definite integral finds the area under a curve by doing the following: $$\int_a^b f(x)dx=F(b)-F(a)$$ And so my question is, how exactly is a definite integral calculated? Why can you simply find the antiderivative of a function to find the area under the function by plugging in the x values into the antiderivative and then subtracting the two results? How are antiderivatives and areas under curves even related?

marked as duplicate by Mark S., Eric Wofsey, Leucippus, Cesareo, Eevee TrainerMar 31 at 1:35

• This relationship is called the "Fundamental Theorem of Calculus", which you may have heard of. – Eric Wofsey Mar 30 at 21:08