What is the physical meaning of an integral?

Question

The derivative $\frac{dy}{dx}$ of a function $y=f(x)$ tells us how has the function $y=f(x)$ changes with the change in $x$ at the point $(x,y)$. What is the physical meaning of the integral of the function $y=f(x)$ i.e., $I(a,b)=\int\limits_{a}^{b} f(x)dx$ except the fact that it represents the area under the curve bounded by $x=a$, $x=b$ and $y=f(x)$?

To be specific the work done under a force, in one-dimension, is given by $\int F(x)dx$. Why should it be called a continuous sum?

How does the area interpretation work out if the function being integrated is a function of several variables?

Answer 1

Here is a nice video by 3Blue1Brown with visuals

Answer 2

To the extent that one interprets a differential form representing the "infinitesimal" variations of a function, the fundamental theorem of calculus can be directly seen as taking the infinitesimal variation and producing the total variation:

$$\int_{x=a}^{x=b} \mathrm{d}f(x) = f(b) - f(a) $$

Answer 3

When integrating the displacement function, call it $f(x),$ you compute the "absement.$ In kinematics, the absement is the measure of sustained displacement. (Wikipedia)