When we start learning calculus, we tend to be given a nice understanding of where derivatives come from, and how the process links to tangents. But how are integrals linked to area?
I found a nice explanation of this here, and I thought I might have a go at rehashing their explanation.
So, say you want to find the area under some curve, from x to (x+dx).
Let’s say you have an area function A, which is defined in such a way that A(x+dx)-A(x) gives the area between x and x+dx:
Area from x to (x+dx) = A(x+dx) – A(x)
Now, this is roughly equivalent to a rectangle of width dx and height f(x). as such, we say:
A(x+dx)-A(x) ≈ f(x)dx
And as such:
(A(x+dx)-A(x))⁄dx ≈ f(x)
Now, the smaller dx is, the more and more accurate this becomes. As dx approaches 0:
limdx→0 (A(x+dx)-A(x))⁄dx = f(x)
Our approximation gets closer and closer to being correct! But you might also notice that the expression on the left is the equivalent of differentiating the function A, so:
limdx→0 (A(x+dx)-A(x))⁄dx = A'(x) = f(x)
So our original function is the derivative of our area function! as such, anti-differentiating (or integrating) our function gives us the function for the area.
This is a really nice, non-rigorous explanation of why integration gives the area.