Is my interpretation of integration correct?

Question

I thought of this when I was thinking about the $dx$ under the integral sign. So we have a function $y=f(x)$. Therefore, $dy/dx=f'(x)$, so $dy=f'(x)dx$. Now the graph of $f$ is split into small $constant$ widths called $dx$. Now we $define$ an operation $"∫"$ such that if I input a small change in $y$ over one of these constant interval of length $dx$, the operation will spit out the y-coordinate at which this change occurred, so $∫dy=y=f(x)$. Therefore, $f(x)=∫f'(x)dx$. Is this interpretation correct? Thanks!

Answer 1

It sounds like you're trying to start out by assuming the Fundamental Theorem, which while not wrong, is not a particularly good place to start. The classic intuitive interpretation of integration approaches this idea a little differently:

You want to know how much a function F(x) changes from a to b, so you sample the function f(x) representing how much F(x) is changing at x at one $x_i^*$ in each interval $[x_i, x_{i+1}]$. Over the interval $[x_i, x_{i+1}]$, F(x) changed by about $f(x_i^*)(x_{i+1}-x_i)$.

You add up all these little changes from $x_0=a$ to $x_n=b$, which it can be shown will give you F(b)-F(a).