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

Question

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?

Answer 1

A very simple explanation that our lecturer gave us:

To find a good estimate of area under a graph, we can break it down into a finite number of regular shapes (triangles, rectangles, etc) and use respective formulas to find the area of each shape then add them all to get an estimate of the total area.

But the issue is that this area might be greater or less than the actual area as regular polygons are formed of straight lines which can go over or below the section of the curve we are trying to estimate the area under.

What one can notice is that, the smaller the sections, the lower the difference between the estimated area and the actual area. So how about we simply split the area under the graph into an infinite number of section with very small sizes for each segment? This will cause the error to approach zero.

Arithmetic addition is used for adding a finite number of quantities. Integration, on the other hand, can be used to add an infinite amount of quantities, thus why integration is used to give an "accurate" measure of area under a graph.

EDIT: It seems I misunderstood your question. I am sorry for that. I thought you were asking about how integration is related to area under the graph.