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.
Table of Content
What are Triple Integrals?
Triple integral is the term given to three continuous solving of integrals represented through three differential integrals. These three integrals are continuously evaluated with their respective variables to attain a final value of the triple integral. The variables and constants in the triple integral can be either be same or different depending on the integral under consideration.
The three integrals involved in Triple Integration are in such a way that the first integral is first evaluated post which the second and third integrals are evaluated.
Note: In Triple Integral, each integrals can be evaluated irrespective of their order and the final result of the integral will be the same value in all circumstances.
Triple Integral Definition
Triple integral refers to the integration of a function that uses three distinct variable and os calculated Triple Integration is done along a three dimensional object that possesses volume.
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Representation of Triple Integrals
Triple integrals can be represented as below:
\iiint kV dV = \int_0^z\int_0^y\int_0^x kV dx dy dz Where x, y and z are three different variables on which integration is performed
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How to Solve Triple Integrals?
Below are the steps which can be followed for solving Triple Integral Problems :
Step 1: The first step for solving problems involving triple integral is to identify what are the variable and their corresponding differentials used in the triple integral equation.
Here, k is a constant.
\iiint k dV = \int_0^z\int_0^y\int_0^x kV dx dy dz So, here variables involved for the above triple integrals are x, y, z represented as dx, dy, dz.
Step 2: Now supposedly the variables are given values, supposedly each of x, y and z is given a lower value of zero and an upper value of 6. So, the equation will look like:
\iiint k dV = \int_0^6\int_0^6\int_0^6 k dx dy dz When values are allotted to variables we will evaluate each of the integral and then put values for each of the variable.
Step 3: In this step, we will evaluate the variable to arrive at a final solution for the integral.
\iiint k dV = \int_0^6\left(\int_0^6\left(\int_0^6 k dx\right) dy\right) dz\\ \Rightarrow \iiint k dV = \int_0^6\left(\int_0^6 k[x]_0^6 dy\right) dz\\ \Rightarrow \iiint k dV = 6k \int_0^6 [y]_0^6 dz\\ \Rightarrow \iiint k dV = 36k\int_0^6 dz \\ \Rightarrow \iiint k dV = 216 k
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.
\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
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,
\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
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
\iiint R f(x,y) dz dy dx ≥ \iiint R g(x,y)dy~dx
\iiint R k f(x, y, z) dz dy dx = k \iiint R f(x, y, z) dz dy dx
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
Application of Triple Integrals
Triple integration can be used in numerous ways to calculate the volume of three dimensional figures. Below listed are the applications of integrals:
- They find their application for three dimensional figures where integral is performed to evaluate the volume. Triple integrals are also referred as volume integrals.
- Supposedly, we have three dimensional variables of x, y, z representing the length, breadth and height of a three dimensional cuboid. This can be calculated through the below formula:
\iiint k dV = \int_0^z \int_0^y\int_0^x k dx dy dz
\iiint k f( x ,y ,z) dx dy dz
- Triple integrals can thus be used for all kind of volume evaluations.
Triple Integrals in Engineering Mathematics
Triple integrals are a fundamental tool in engineering mathematics, used extensively in fields like fluid dynamics, thermodynamics, and electromagnetism. They enable engineers and mathematicians to calculate quantities that are distributed across three-dimensional spaces. Here's the uses of triple integrals in real life:
- Volume Calculations
- Mass and Density Calculations
- Center of Mass
- Moment of Inertia
- Fluid Dynamics and Heat Transfer
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Solved Examples on Triple Integrals
Example 1. Evaluate the triple integral problem:
Solution:
\iiint k dV =\int_0^{z=12}\int_0^{y=12}\int_0^{x=12} k dx dy dz ⇒
\iiint k dV =\int_0^{z=12}\int_0^{y=12} ⇒ k [x] 0 x=12 dy dz⇒
\iiint k dV = \int_0^{z=12}\int_0^{y=12} k [x]0 x=12 dy dz⇒
\iiint k dV = \int_0^{z=12} 12k[y]0 y=12 dz⇒
\iiint k dV = 144k [z]0 z=12⇒
\iiint k dV = 1728 k
Example 2. Evaluate the triple integral problem
Solution:
\int_0^{z=8}\int_0^{y=6}\int_0^{x=4} k dx dy dz =
\int_0^{z=8}\int_0^{y=6} k [x]0 x=4 dy dz=
\int_0^{z=8} 4k[y]0 y=6 dz= 24k [z]0 z=8
= 192 k
Example 3. Evaluate the triple integral problem
Solution:
\int_0^{z=3}\int_0^{y=2}\int_0^{x=4} 4x dx dy dz =
\int_0^{z=3}\int_0^{y=2} 4k [x2/2]0 x=4 dy dz=
\int_0^{z=3} 4k[8][y]0 y=2 dz= 64k [z]0 z=3
= 192 k
Example 4. Evaluate the triple integral problem
Solution:
\int_0^{z=10}\int_0^{y=12}\int_0^{x=5} k dx dy dz =
\int_0^{z=10}\int_0^{y=12} [x]0 5 dy dz=
\int_0^{z=10} [y]0 12 5k dz= [z]0 10 60k
= 600k
Example 5. Evaluate the triple integral problem
Solution:
\int_0^{z=18}\int_0^{y=9}\int_0^{x=3} k dx dy dz =
\int_0^{z=18}\int_0^{y=9} [x]0 x=3 dy dz=
\int_0^{z=18} [y]0 y=9 3k dz= [z] 0 z=18 27k
= 486k
Practice Questions on Triple Integrals
Q1. Solve:
Q2. Solve:
Q3. Solve:
Q4. Solve:
Q5. Solve:
Summary
Triple integrals extend the concept of integration to three dimensions, allowing us to calculate volumes, masses, and other properties of three-dimensional regions. Denoted as ∭ f(x,y,z) dV, they involve integrating a function f(x,y,z) over a volume element dV. The choice of coordinate system (Cartesian, cylindrical, or spherical) affects the form of dV and can simplify calculations based on the problem's symmetry. Key applications include volume calculation, mass determination for objects with varying density, and field calculations in physics. Evaluation typically involves iterated integrals, where the order of integration can be crucial for simplification. Challenges often arise in visualizing the region of integration, setting appropriate limits, and choosing the most efficient coordinate system. Mastery of triple integrals requires practice in visualization, limit-setting, and coordinate transformation techniques.
