Applications of triple integration in engineering mathematics 1

1. PRESENTATION
p r e s e n t e d b y
S . S A I S R I N I V A S - 2 2 H 5 1 A 0 4 B 7
S A D I T H Y A - 2 2 H 5 1 A 0 4 B 8
S . G H I T H E S H - 2 2 H 5 1 A 0 4 B 9

2. Triple integral is a concept in multivariable calculus that allows you
to calculate the volume of a solid region in three-dimensional
space. It's an extension of the double integral, which is used to
calculate the area of a two-dimensional region.

3. Formule for Triple
Integrals
• A triple integral can be expressed in Cartesian, cylindrical, or
spherical coordinates and its limits can be defined using various
shapes such as cubes, cylinders, and spheres. The value of the
triple integral represents the amount of space that the solid
region occupies in three-dimensional space.
• The basic formula for a triple integral in Cartesian coordinates is
∫∫∫f(x,y,z)dV
• where f(x,y,z) is a continuous function that defines the density of
the solid region and dV is the volume element.

4. Example problem
Example 1
Example 2

6. 1. Calculation of volume: Triple integrals can be used to calculate the
volume of a solid object, by dividing the object into infinitesimal
pieces and summing their volumes.
2. Mass and density distribution: Triple integrals can be used to
determine the mass and density distribution of an object, which is
important in fields such as
3.Fluid dynamics: Triple integrals can be used to analyze the flow of
fluids and calculate properties such as velocity and pressure. This is
important in fields such as

7. Histor
y
• The concept of integration can be traced back to the ancient Greeks, who were
interested in finding the area under curves. However, it wasn't until the
development of calculus in the 17th century by mathematicians such as Isaac
Newton and Gottfried Leibniz that the modern concept of integration was
developed.
• Triple integrals, which involve integrating over three dimensions, were developed as part
of the wider field of multivariable calculus, which studies functions of more than one
variable. The concept of integrating over multiple dimensions was first developed by
mathematicians such as James Clerk Maxwell and Hermann Grassmann in the 19th
century.