Properties of Triple Integration
Some of the properties of Triple Integration are:
- Linearity
- Additivity
- Monotonicity
- Divergence Theorem
Note: Divergece Theorem is not a property of Triple Integration in literal sense, but as it involves the triple integral or volume integral. Thus we can consider this as property.
Linearity
This property denotes triple integrals to give the same result under same limits when addition/subtraction is performed collectively on the triple integral and when addition/subtraction is performed on individual units of triple integral variable terms.
[Tex]\iiint R [f(x,y,z) \pm g(x,y,z)] dV = \iiint R f(x,y,z) dV \pm \iiint R g(x,y,z) dV [/Tex]
Additivity
This property denotes triple integrals to give the same result under same limits when evaluated as a single unit or split into multiple units. Here, R is region given as a union of S and T where S and T are disjoint partition of R i.e. S and T units have nothing in common.
If R = S ⋃ T, S ⋂ T = Φ then,
[Tex] \iiint R f(x,y,z) dz dy dx = \iiint S f(x,y,z)dz dy dx + \iiint T f(x,y) dz dy dx [/Tex]
Monotonicity
This property denotes triple integrals to be fetching the same results for the same limits irrespective the variable is evaluated as a part of triple integral or it is outside the triple integral evaluation.
If f(x, y, z) ≥ g(x, y), then [Tex]\iiint R f(x,y) dz dy dx ≥ \iiint R g(x,y)dy~dx [/Tex]
[Tex]\iiint R k f(x, y, z) dz dy dx = k \iiint R f(x, y, z) dz dy dx [/Tex]
Divergence Theorem
The theorem mentions of the normal component of a vector point function supposedly take it as F over a closed surface say ‘S’ is the volume integral of divergence of ‘F’ taken over volume ‘V’ enclosed by the closed surface S.
It is denoted as below
[Tex]\iiint_V ▽\vec F. dV = \iint_s \vec F. \vec n. dS [/Tex]
Triple Integrals
Triple Integrals: Integrals are an essential part of the mathematical world and hold great significance in today’s world. There are different types of integrals and each has its importance in mathematics. Some of these different types of integrals in mathematics are linear integrals, double integrals, triple integrals, etc.
Triple Integral is one of the types of multi integral of a function that involves three variables. Triple Integral in Calculus is the integration involving volume, hence it is also called Volume Integral and the process of calculating Triple Integral is called Triple Integration.
In this article, we will discuss triple integrals in detail along with their examples and representation and steps to solve multiple triple integral problems.
Read in Detail: Integrals
Table of Content
- What are Triple Integrals?
- Triple Integral Definition
- Representation of Triple Integrals
- How to Solve Triple Integrals?
- Properties of Triple Integration
- Linearity
- Additivity
- Monotonicity
- Divergence Theorem
- Application of Triple Integrals
- Triple Integrals in Engineering Mathematics
- Solved Examples on Triple Integrals
- Practice Questions on Triple Integrals