# Why is the area under a curve the integral?

## Main Question or Discussion Point

here is a geometric proof, similar to the one in my textbook (copied from Aryabhata, from http://math.stackexchange.com/questions/15294/why-is-the-area-under-a-curve-the-integral) : Is this saying: that the A' equals the function. Which is implying, that the integration of A equals F (where F is the integral of f)?

thanks

Fredrik
Staff Emeritus
Gold Member
You don't prove that the integral of a function is the area under its graph. What you do is to extend the definition of the "area" (which is initially only defined for rectangles) to regions of shapes like this, by saying that the area of such a region is the integral of the function. This is just a choice to use the word "area" in these situations, nothing more. So it can't be proved.

What you can do is to explain why the word "area" was chosen for this concept.

The argument you posted says that A'=f. By the fundamental theorem of calculus, this implies that for all real numbers a,b,
$$\int_a^b f(x)dx =\int_a^b A'(x)dx =A(b)-A(a).$$

Last edited:
Fredrik
Staff Emeritus