What is the meaning of the integral of volume?

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SUMMARY

The integral of volume, computed as (πr^4)/3, represents a four-dimensional volume associated with a solid sphere. This concept extends the understanding of integrals beyond traditional applications of area and volume, suggesting a relationship to higher dimensions. The discussion emphasizes that while integrals can yield numerical results, their deeper meanings, particularly in four-dimensional contexts, require abstract visualization and interpretation. The derivative of volume leads back to the surface area, reinforcing the interconnectedness of these mathematical concepts.

PREREQUISITES
  • Understanding of basic calculus concepts, including integrals and derivatives.
  • Familiarity with the geometric properties of spheres and their equations.
  • Knowledge of dimensional analysis, particularly in relation to volume and area.
  • Conceptual grasp of higher-dimensional spaces and their implications in mathematics.
NEXT STEPS
  • Explore the implications of integrals in higher dimensions, focusing on four-dimensional geometry.
  • Study the relationship between volume and surface area through calculus, particularly in spherical coordinates.
  • Investigate the applications of integrals in physics, especially in contexts involving density and material properties.
  • Learn about the visualization techniques for higher-dimensional objects and their mathematical representations.
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Mathematicians, physics students, and educators interested in advanced calculus concepts, particularly those exploring the implications of integrals in higher dimensions and their applications in various fields.

mljoslinak
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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.
 
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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.
 
"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.
 

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