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.

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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.

Representation of Triple Integrals

Triple integrals can be represented as below:

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.

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:

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.

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.

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,

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

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:

• 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

Solved Examples on Triple Integrals

Example 1. Evaluate the triple integral problem:

Solution:

⇒ ⇒ k [x] ₀ ^x=12 dy dz

⇒ k [x]₀ ^x=12 dy dz

⇒ 12k[y]₀ ^y=12 dz

⇒ = 144k [z]₀ ^z=12

⇒ k dV = 1728 k

Example 2. Evaluate the triple integral problem

Solution:

= k [x]₀ ^x=4 dy dz

= 4k[y]₀ ^y=6 dz

= 24k [z]₀ ^z=8

= 192 k

Example 3. Evaluate the triple integral problem

Solution:

= 4k [x²/2]₀ ^x=4 dy dz

= 4k[8][y]₀ ^y=2 dz

= 64k [z]₀ ^z=3

= 192 k

Example 4. Evaluate the triple integral problem

Solution:

= [x]₀ ⁵ dy dz

= [y]₀ ¹² 5k dz

= [z]₀ ¹⁰ 60k

= 600k

Example 5. Evaluate the triple integral problem

Solution:

= [x]₀ ^x=3 dy dz

= [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:

FAQs on Triple Integrals

How many Integrations are involved in Triple Integrals?

As the name suggests, we perform the integration thrice here.

Which Figures are Evaluated with the help of Triple Integrals?

Three dimensional shapes are evaluated for their volumes using triple integrals.

How do we differentiate between Double Integration and Triple Integration?

In double integration, we perform the integration twice and there are utmost two differential variables involved whereas in triple integration we perform the integration thrice and there are utmost three distinct differential variables involved.

What is the Order of Integration in Triple Integration?

We can evaluate any integral first in triple integration, there is no specific order to be followed here.

Can Triple Integrals have Constants present along with Variables?

Yes, triple integrals can have constants present along with them in the triple integration mathematical problem.

Does the final evaluation answer of triple integral change on the changing the integral evaluation sequence?

No, the answer of the triple integral remains the same irrespective which integral from the triple integral is evaluated first.

What is Meaning of Triple Integration?

Triple integration as the name suggests involves three distinct variable integrations in it’s evaluation for the triple integral problem.

What is the Formula of Triple Integral?

Below is the formula of triple integrals:

What is the Triple Integral with Limits?

Triple integral with limits refer to the different values each of the variable in the triple integral term can have values from. Once variables are

Can Triple Integral be Negative?

Yes, triple integrals can be negative as well as positive. There is no restriction on triple integrals for negative sign.