Physical significance of integral of volume?

Question

If we take the function f(x)=1 as an example, it is a 1D line.

Its integral is represented as a 2D area.

Using that area as a cross-section of a prism, the integral of the area is 3D volume.

So, let's imagine a sphere. This sphere begins with radius 2m, and remains spherical, but its radius increases 1m every second.

What is the physical significance of the integral of the spheres' volumes with respect to time? A 4D measurement? Is this a way to measure time itself, or something else?

If it does not follow the pattern, why not?

-IMP ;) :)

Answer 1

I'm not sure what you're thinking.  The integral of a function of one variable isn't a line or an area, it's simply an integral.  If you're renting helium to fill a balloon, then the integral of its volume over time is a measure of your helium cost as a function of time.