What is the Integral of a Volume?

• BrianJester
In summary, the relationship between length, area, and volume in geometry extends to the 4th dimension, where the corresponding term is "hypervolume." This concept can be extended to higher dimensions, with a corresponding "m-volume" or "m-measure" for any integer m. It is important to note that these terms are specific to geometry and should not be confused with the physics terms of "speed," "acceleration," "Jerk," and "Jounce (or snap)."
BrianJester
We know the relationship of:

speed > acceleration > Jerk > Jounce (or snap)

What is the similar relation ship of:

length > area > volume > ?

My question is: what comes after volume in this pattern?

-4th dimensional volume?

Also how do I refer to this?

-The integral of volume?

Brian Jester

Talking of "length", "area" , "volume" means that we are in the field of geometry, respectively in 1, 2, 3 dimensions spaces. So, extending the concept implies to go to the 4th dimension and the corresponding name is "hypervolume", in fact 4D.hypervolume. Then we have 5-D.hypervolume, ..., n-D.hypervolume.
The formulas corresponding to hyper-sphere, hyper-cone, hyper-sphreical cap in n-dimensions spaces are available in the paper "Le problème de l'hyper-chèvre" :
http://www.scribd.com/JJacquelin/documents

The "integral of a volume" doesn't mean anything. You integrate functions, not geometric quantities. But that just says that "integral" is the wrong word.

In three dimensional geometry there is nothing "beyond" volume. Mathematically, you can, however, talk about n-dimensional spaces for any integer n. In that case, for any $m\le n$, we can talk about the "m-volume" or "m-measure". "Length" is "1-volume" or "1-measure", "area" is "2-volume" or "2-measure", etc.

By the way, "speed > acceleration > Jerk > Jounce (or snap)" are physics terms, not mathematics.

JJacquelin,

Thank you for the explanation, after revisiting the hypercube, I can conceptualize a hyper volume.

Brian Jester

1. What is the definition of "integral of a volume"?

The integral of a volume is a mathematical concept used to find the total volume of a three-dimensional shape or region. It involves breaking the shape or region into infinitesimally small pieces and summing their volumes using integration.

2. How is the integral of a volume different from the integral of a function?

The integral of a volume is used to find the total volume of a three-dimensional shape, while the integral of a function is used to find the area under the curve of a two-dimensional function. In other words, the integral of a volume is a three-dimensional concept, while the integral of a function is a two-dimensional concept.

3. Can the integral of a volume have negative values?

No, the integral of a volume always yields a positive value as it represents the total volume of a shape or region. Negative values would not make sense in the context of volume.

4. What is the relationship between the integral of a volume and the derivative of a volume?

The integral of a volume and the derivative of a volume are inverse operations. This means that if the integral of a volume is used to find the total volume of a shape, the derivative of that volume can be used to find the rate of change of the volume with respect to another variable.

5. How is the integral of a volume used in real-life applications?

The integral of a volume has many practical applications in fields such as engineering, physics, and architecture. It is used to calculate volumes of objects or structures, such as determining the amount of liquid in a tank, the volume of a building, or the capacity of a container. It is also used in fluid mechanics to calculate the flow rate of a fluid through a given volume.

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