Calculus III
Section 14.4: Applications of Double Integrals
Density and Mass
Given a lamina (thin plate) occupies a region D in
the xy-plane and its density (units of mass/unit area)
at point (x, y) in D is given by ρ(x, y), where ρ is a
continuous function on D, to find the total mass m:
∬ ( )
A similar application would be for electric charge
over a region D. If the charge density (units of
charge/unit area) is given by σ(x, y) at a point (x, y)
in D, then the total charge Q is:
∬ ( )
Example 1: Charge is distributed over the triangular
region D bounded by y = 1-x, x = 1, y = 1 and the
char2e density at (x, y) is σ(x, y) = xy, measured in
C/m . Find the total charge Moments and Center of Mass
Recall: The moment of a particle about an axis is
the product of its mass and its directed distance from
the axis.
So to get the moment of a lamina about the x-axis:
∬ ( )
Similarly, the moment of a lamina about the y-axis:
∬ ( )
The center of mass is the point (x*, y*) such that
mx* = M ynd my* = M x In other words, the
lamina behaves as if its entire mass is concentrated
at its center of mass. Therefore:
x* = (1/m)My= ∬ ( ) and
y* = (1/m)Mx= ∬ ( ) Example 2: Find the mass and center of mass of a
triangular lamina with vertices (0, 0), (1, 0) and (0,
2) if the density function is ρ(x, y) = 1 + 3x + y
Example 3: The density at any point on a
semicircular lamina is proportional to the distance
from the center of the circle. Find the center of mass
of the lamina.
Moment of Inertia
Also known as second moment
Is the product of the mass of a particle and the
square of its distance to the axis
So to find the moment of inertia of a lamina about
the x-axis or y-axis:
∬ ( )
∬ ( )
We can also find the moment of inertia about the
origin (called the polar moment of inertia) by: ∬( ) ( )
Example 4: Find the moments of inertia Ix, y 0 I of a
homogeneous disk D with density ρ(x, y) = ρ, center
at the origin, and radius a
Probability
We can now consider a pair of continuous random
variables X and Y such as lifetimes of two
components of a machine or the height/weight of
randomly selected male students
The joint density function of X and Y is a function
f of two variables such that
( ) ∬ ( )

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