Why is the volume integral of zero equal to zero?

In summary, the conversation discusses the difference between volume integrals and indefinite integration, with the conclusion that the volume integral of zero is always equal to zero.
  • #1
tomwilliam2
117
2
If the integral of zero is a constant, then why is the volume integral of zero just zero?
 
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  • #2
You're addressing two types of integration. <Volume integrals> can be expressed as interated definite integration, while <integral of 0 is a constant> means that you're speaking of indefinite integration.
 
  • #3
tomwilliam2 said:
If the integral of zero is a constant, then why is the volume integral of zero just zero?

Consider the expression ##\displaystyle \iiint\limits_{V}0 \, dV##, which I believe you mean by "volume integral". Since 0 is, itself, a constant, we can pull it out front, getting ##\displaystyle 0\iiint\limits_{V} \, dV = 0V = 0##
 
  • #4
Thanks, both of you.
 
  • #5


The volume integral of zero being equal to zero is a result of the fundamental theorem of calculus, which states that the integral of a constant function is equal to the constant multiplied by the interval of integration. In this case, the interval of integration is the volume of the region being integrated, and since the function being integrated is constantly zero, the result is also zero. This can be understood intuitively by considering the definition of a volume integral, which is the sum of infinitely small volumes over a given region. Since the function being integrated is zero, the contribution of each infinitely small volume is also zero, resulting in a total integral of zero. Therefore, the volume integral of zero is just zero due to the nature of the function being integrated, and is not dependent on the interval of integration.
 

1. What is a volume integral?

A volume integral is a mathematical concept used in calculus and physics to calculate the total amount of a quantity within a three-dimensional region. It involves integrating a function over a volume, which can be thought of as finding the sum of infinitely small volumes within the given region.

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

A volume integral is a type of triple integral, which involves integrating over three variables (x, y, and z) instead of just one variable like in a regular integral. Additionally, a volume integral is used to calculate the total amount of a quantity within a three-dimensional region, while a regular integral is used to calculate the area under a curve or the length of a line.

3. What are some common applications of volume integrals?

Volume integrals are commonly used in physics to calculate the mass, charge, or energy within a given region. They are also used in engineering to calculate properties such as moment of inertia and center of mass. In mathematics, volume integrals are used to calculate the volume of irregular shapes and to solve problems involving fluid flow and electric fields.

4. How do you set up a volume integral?

To set up a volume integral, you need to determine the limits of integration for each variable (x, y, and z) and the function to be integrated. This is typically done by first visualizing the region and drawing a diagram, then setting up the integral based on the boundaries of the region. You may also need to convert the function to be integrated into the appropriate coordinate system (cartesian, cylindrical, or spherical) depending on the shape of the region.

5. Are there any tricks or shortcuts for solving volume integrals?

There are some techniques that can make solving volume integrals easier, such as using symmetry to simplify the bounds of integration or changing the order of integration. However, in general, volume integrals can be quite complex and often require a lot of algebraic manipulation. It is important to have a strong understanding of calculus and the properties of integrals in order to effectively solve volume integral problems.

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