Meaning of volume integral.

[mljoslinak]

I have been thinking about the meaning of integrals and derivatives. For instance, the area of a sphere is 4 pi r^2. I can get that. The derivative of the area is 8 pi r or 4 times the circumference of the sphere. The derivative of this is just 8 pi. I can kind of understand that too. Then you go to 0 if you differentiate again. I'm fine with that.

Now we go the other way. The integral of area is volume or 4/3 pi r^3. I can understand that too. Here's the catch. What is the meaning of the integral of volume? I can compute it easily to be (pi r^4)/3, but what does that mean?

I wondered about density, but that should be dependent on the material. I also wondered about it being the time in the sphere since that is the fourth dimension.

[CompuChip]

You already noted that the derivative of the three dimensional volume of the sphere is the two dimensional volume (aka area) of its surface.

So 1/3 pi r^4 would be the (four dimensional) volume of an object (which you can only visualise in four spatial dimensions, and I don't think many people can easily do that) of which the "surface" would be a solid sphere.

[HallsofIvy]

"Area" and "volume" are possible applications of integrals. It would be a mistake to think that they are, in any important sense, the "meaning" of the integral.