Question

Whether complex integral represent area under curve?

Answer 1

If you’ve taken calculus, then at some point you learned that to find the area under a function (generally written ) you need to find the anti-derivative of that function.  The most natural response to these types of theorems is “wait… what?… why?”.

This theorem is so important and widely used that it’s called the “fundamental theorem of calculus”, and it ties together the integral (area under a function) with the antiderivative (opposite of the derivative) so tightly that the two words are essentially interchangeable.  However, there are some mathematicians who may take issue with mixing up the two terms.

It comes back (in a roundabout way) to the fact that the derivative of a function is the slope of that function or the “rate of change”.  In what follows “f” is a function, and “F” is its anti-derivative (that is: F’ = f).

Intuitively: Say you’ve got a function f(x), and the area under f(x) (up to some value x) is given by A(x).

Then the statement “the area, A, is given by the anti-derivative of f” is equivalent to “the derivative of A is given by f”.

In other words, the rate at which the area increases (as you slide x to the right) is given by the height, f(x).