ILOCOS SUR POLYTECHNIC STATE COLLEGE
Tagudin Campus
MODULE
LESSON APPLICATION OF TRIPLE
1 INTEGRAL
Triple integrals are an advanced concept in calculus, widely used to compute quantities
over three-dimensional spaces. Unlike single or double integrals, which deal with one-dimensional
curves or two-dimensional areas, triple integrals extend this idea into three dimensions, allowing
us to calculate properties like volume, mass, or energy in complex three-dimensional regions.
These integrals operate by summing infinitesimally small contributions of a function (f(x, y, z)) over
a given region (R) in space.
The versatility of triple integrals lies in their ability to adapt to different coordinate systems,
including Cartesian, cylindrical, and spherical coordinates. This adaptability makes them
particularly useful in solving problems involving irregularly shaped objects, complex density
distributions, or scenarios where symmetry simplifies computations. Their application spans
various disciplines such as physics, engineering, biology, and even economics, highlighting their
importance in theoretical and practical contexts.
In essence, triple integrals are a fundamental mathematical tool that enables us to quantify
and analyze physical phenomena and geometric properties within three-dimensional spaces.
Their applications are vast and varied, ranging from determining the mass of a solid with varying
density to calculating electric fields, fluid flow, and even probabilities in multidimensional spaces.
By setting appropriate integration limits and understanding the geometry of the region (R), triple
integrals provide precise and powerful solutions to complex real-world problems.
1. Volume Calculation
Triple integrals are commonly used to find the volume of a three-dimensional region R.
Where dV is the volume element, often written in Cartesian (dx dy dz), cylindrical (r dr dθ dz)), or
spherical coordinates (ρ2sin2 ϕ dρ dϕ dθ).
2. Mass of a Solid
For a solid with varying density ρ(x,y,z), the mass can be calculated as:
❑
M =∭R p ( x , y , z ) dV
Instructor: Dr. Jonathan Dela Cruz
ILOCOS SUR POLYTECHNIC STATE COLLEGE
Tagudin Campus
MODULE
3. Center of Mass and Moment of Inertia
● Center of Mass ( x , y , z )
❑ ❑ ❑
∭ xp( x , y , z )dV ∭ yp (x , y , z)dV ∭ zp (x , y , z)dV
x= R y= R z= R
M M M
● Moment of Inertia:
❑ 2 2
∭ R ( x + y ) p(x , y , z )dV
I=
M
4. Electric and Gravitational Fields
● Electric Field: The electric potential V due to a charge distribution ρ(x,y,z) can be
computed using:
' '
1 p(x , y , z ')
V (x , y , x )= ∭❑
4 π ϵθ R
√❑
● Gravitational Potential: Similar integrals apply for gravitational fields, replacing the
constants.
5. Fluid Dynamics
Triple integrals calculate fluid quantities such as mass flow rate or center of mass within a
region, considering density and velocity.
6. Heat Transfer and Energy
Triple integrals are used to determine the total energy or heat within a volume. For example, the
total thermal energy of a region is given by:
❑
E=∭ R cp ( x , y , z) T ( x , y , z )dV
Where ccc is the specific heat capacity, p is the density, and T (x , y , z) is the temperature.
7. Probability in 3D
In probability, triple integrals evaluate the total probability within a 3D region using a probability
density function f(x,y,z):
Instructor: Dr. Jonathan Dela Cruz
ILOCOS SUR POLYTECHNIC STATE COLLEGE
Tagudin Campus
MODULE
❑
P=∭ R f ( x , y , z ) dV
Instructor: Dr. Jonathan Dela Cruz
ILOCOS SUR POLYTECHNIC STATE COLLEGE
Tagudin Campus
MODULE
Instructor: Dr. Jonathan Dela Cruz
ILOCOS SUR POLYTECHNIC STATE COLLEGE
Tagudin Campus
MODULE
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Instructor: Dr. Jonathan Dela Cruz
ILOCOS SUR POLYTECHNIC STATE COLLEGE
Tagudin Campus
MODULE
References
https://math24.net/calculation-volumes-triple-integrals.html
Instructor: Dr. Jonathan Dela Cruz
