Area under curve fundamentals

I'm reading Stroud's Engineering Mathematics. A small excerpt from Stroud's book follows: Area under the curve_part1

Area under the curve_part2

I'm a software engineer and I know how to integrate but I'm currently looking into its fundamentals. How would you define $dA_x$ [Refer image 2]? I know dy/dx is explained as the slope of the tangent to a point on a curve - I get this, but not $dA_x$. I find it odd, how the explanation very conveniently picks $\delta x$ as the area of the strip and as the error drops down to 0, $\delta x$ tends to 0 and hence becomes $dA_x$. What have I got wrong?

Conceptually, $dy/dx$ is the limit of $\delta y/\delta x$ as $\delta x\to 0$; this happens to give the slope of the tangent to the curve.
Similarly, $dA_x/dx$ is just the limit of $\delta A_x/\delta x$ as $\delta x\to 0$. You can (if you want) think of this as the slope of the tangent to the curve $A(x)$ where $A(x)$ is the area under the curve from some fixed point $a$ to $x$. So the two concepts are really the same.