Transcript Document 7827757

Section 16.5
Applications of Double
Integrals
LAMINAS AND DENSITY
A lamina is a flat sheet (or plate) that is so thin as to be
considered two-dimensional.
Suppose the lamina occupies a region D of the xyplane and its density (in units of mass per area) at a
point (x, y) in D is given by ρ(x, y), where ρ is a
continuous function on D. This means that
m
 ( x, y )  lim
A
where Δm and ΔA are the mass and area of a small
rectangle that contains (x, y) and the limit is taken as the
dimensions of the rectangle approach 0.
MASS OF A LAMINA
To find the mass of a lamina, we partition D into small
*
*
(
x
,
y
rectangles Rij of the same size. Pick a point ij ij )
in Rij. The mass of Rij is approximately  ( xij* , yij* ) A( Rij ),
and the total mass of the lamina is approximately
k
l
m    ( xij* , yij* ) A.
i 1 j 1
The actual mass is obtained by taking the limit of the
above expression as both k and l approach zero. That
is,
k
l
m  lim   ( xij* , yij* ) A    ( x, y) dA.
k ,l  
i 1 j 1
D
EXAMPLE
Find the mass of the lamina bounded by the
triangle with vertices (0, 1), (0, 3) and (2, 3) and
whose density is given by ρ(x, y) = 2x + y,
measured in g/cm2.
MOMENTS
The moment of a point about an axis is the product of
its mass and its distance from the axis.
To find the moments of a lamina about the x- and yaxes, we partition D into small rectangles and assume
the entire mass of each subrectangle is concentrated at
an interior point. Then the moment of Rij about the xaxis is given by


(mass )( yij* )   ( xij* , yij* )A yij*
and the moment of Rk about the y-axis is given by


(mass )( xij* )   ( xij* , yij* )A xij*
MOMENTS (CONCLUDED)
The moment about the x-axis of the entire
lamina is
M x  lim
m,n  
m
n
 y
i 1 j 1
*
ij
 ( x , y ) A   y  ( x, y) dA
*
ij
*
ij
D
The moment about the y-axis of the entire
lamina is
M y  lim
m, n  
m
n
*
*
*
x

(
x
,
y
 ij ij ij ) A   x  ( x, y) dA
i 1 j 1
D
CENTER OF MASS
The center of mass of a lamina is the “balance point.”
That is, the place where you could balance the lamina on
a “pencil point.” The coordinates (x, y) of the center of
mass of a lamina occupying the region D and having
density function ρ(x, y) is
My
1
x
  x  ( x, y ) dA
m
m D
Mx 1
y
  y  ( x, y ) dA
m m D
where the mass m is given by
m    ( x, y) dA
D
EXAMPLES
1. Find the mass and center of mass of the
triangular lamina bounded by the x-axis and
the lines x = 1 and y = 2x if the density
function is ρ(x, y) = 6x + 6y + 6.
2. Find the center of mass of the lamina in the
shape of a quarter-circle of radius a whose
density is proportional to the distance from the
center of the circle.
MOMENTS OF INERTIA
The moment of inertia (also called the second
moment) of a particle of mass m about an axis is
defined to be mr2, where r is the distance from
the particle to the axis. We extend this concept to
a lamina with density function ρ(x, y) and
occupying region D as we did for ordinary
moments.
MOMENTS OF INERTIA
CONCLUDED
The moment of inertia about the x-axis is
I x   y  ( x, y ) dA
2
D
The moment of inertia about the y-axis is
I y   x  ( x, y ) dA
2
D
The moment of inertia about the origin (or polar moment)
is
2
2
I 0   ( x  y )  ( x, y ) dA  I x  I y
D
EXAMPLE
Find the moments of inertia Ix, Iy, and I0 of the
lamina bounded by y = 0, x = 0, and y = 4 − x2
where the density is given by ρ(x, y) = 2y.