# 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