Why do integrals work?

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.



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