Applications of Double Integrals

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Section 16.5

• Applications of Double Integrals

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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 xy-plane
and its density (in units of mass per area) at a
point (x, y) in D is given by ?(x, y), where ?
is a continuous function on D. This means that
where ?m and ?A 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.
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MASS OF A LAMINA
To find the mass of a lamina, we partition D into
small rectangles Rij of the same size. Pick a
point in Rij. The mass of Roj is
approximately and the total mass of the lamina is
approximately
The actual mass is obtained by taking the limit
of the above expression as both k and l approach
zero. That is,
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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 ?(x, y) 2x y,
measured in g/cm2.
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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 y-axes, 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 x-axis
is given by and the moment of Rk about the
y-axis is given by
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MOMENTS (CONCLUDED)
The moment about the x-axis of the entire lamina
is
The moment about the y-axis of the entire lamina
is
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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 ?(x, y) is where the mass m is given
by
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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 ?(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.
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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 ?(x, y) and
occupying region D as we did for ordinary moments.
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MOMENTS OF INERTIA CONCLUDED
The moment of inertia about the x-axis is
The moment of inertia about the y-axis is
The moment of inertia about the origin (or polar
moment) is
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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 ?(x, y) 2y.