I apologize in advance for such a naïve question, but I am learning about integrals right now, and something keeps gnawing at me!
So if we take the definite integral of a function $f'(x)$ over the interval say $[2,5]$, what we are really doing (and how I like to think of it) is summing all of the outputs of the antiderivative function $f(x)$ over the interval $[2,5]$ where every "item" added to the "set of items to sum" is the output from $f(x)$ with an infinitesimally small increase in the previous $x$.
So in my head I see taking the integral as summing the values in the region highlighted purple on $f(x)$ for:
And not as the area under the curve of the derivative function $f'(x)$:
I see this as the "correct" way personally because I am unable to make the connection of what the "area under the curve" really is, besides empty space that our function does not interact with.
Edit for @Dheeraj: In the images below you can see my miss-understanding, they are there to show my previous wrong thinking. I thought that the rectangles would continue splitting until there were no more x values left, and converge to only a y values. But this would not make sense because every time a rectangle gets "smaller" another rectangle appears, it is not as if there are a finite number of rectangles whos widths continue getting smaller. New rectangles are infinitely constructed, which means new y values are constructed too. Also note that they are split up, but it goes from left to right.
Added image that better shows my misunderstanding:
Best Answer
Your definition is Arc Length. If $\int$ is the area under the curve, the arc length of $f(x)$ from $x=x_0$ to $x=x_1$ would be $\int_{x_0}^{x_1} \sqrt{1+(f'(x))^2}\text{ dx}$ where $f'(x)$ is the derivative of $f$ evaluated at each point $x_0 \le x \le x_1$. So essentially, the arc length of the curve $f(x)$ is secretly equal to the area under a transformed version of $f'$.
Area under curve, when the curve is lying in $[-1, +1]$, is a richer class than arc length since arc-length only increases, and is always non-negative. Whereas area under a curve can go negative if your $f$ is negative in enough mass to outweigh the positive mass encountered that far in your definite integral.. Hope that clears it up a bit, basically area under a curve has to be defined before one is able to comprehend the arc length, and the arc length is a specialized subset of all possible areas under curves, so it is not enough to consider only arc lengths.