What is the meaning of the integral of volume?

In summary, The conversation discusses the meaning of integrals and derivatives in relation to the area and volume of a sphere. It is noted that the derivative of the area is 4 times the circumference of the sphere, and the derivative of this is 8 pi. The integral of the area is the volume of the sphere, and the integral of the volume is the four dimensional volume of an object. However, it is stated that "area" and "volume" are only possible applications of integrals and are not the true meaning of the integral.
  • #1
mljoslinak
3
0
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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  • #2
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.
 
  • #3
"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.
 

1. What is the definition of a volume integral?

A volume integral, also known as a triple integral, is a mathematical tool used to calculate the volume of a three-dimensional object or region in space. It involves integrating a function over a three-dimensional domain.

2. How is a volume integral different from a surface integral?

A volume integral involves integrating a function over a three-dimensional region, while a surface integral involves integrating a function over a two-dimensional surface. In other words, a volume integral calculates the volume of a solid, while a surface integral calculates the area of a surface.

3. What is the purpose of using a volume integral in scientific research?

A volume integral is used in scientific research to calculate physical quantities such as mass, density, and electric charge, which are important in various fields of study such as physics, engineering, and mathematics. It is also used in solving differential equations and modeling physical phenomena.

4. How is a volume integral evaluated?

A volume integral is evaluated by dividing the three-dimensional region into smaller, simpler shapes such as cubes or rectangular prisms. The function is then integrated over each of these shapes, and the results are summed together to obtain the final value.

5. What are some real-life applications of volume integrals?

Volume integrals have various real-life applications, such as calculating the mass of an object with varying density, determining the center of mass of a solid object, and finding the electric flux through a three-dimensional region. They are also used in fluid mechanics to calculate fluid flow rates and in thermodynamics to determine the heat transfer in a three-dimensional system.

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