13.6 Applications of Double Integrals
In this section, we explore physical applications such as computing mass, center of
mass, and electric charge.
Total Mass
Suppose the lamina (thin plate) occupies a region D of the xy -plane and its density
(in units of mass per unit area) at a point (x, y) in D is given by ρ(x, y). Then the
total mass of the lamina is given by
¨
m=
ρ(x, y) dA
D
, (x̄, ȳ), of the lamina with density ρ(x, y) that occupies the region
D density ρ(x, y) is dened as
Center of mass
˜
x̄ = D˜
D
xρ(x, y) dA
ρ(x, y) dA
˜
ȳ = D˜
yρ(x, y) dA
ρ(x, y) dA
D
The physical signicance is that the lamina behaves as if its entire mass is concentrated at its center of mass. Thus the lamina balances horizontally when supported at
its center of mass.
Example 1. Find the center of mass of the lamina that occupies the region
D = { (x, y) | x2 + y 2 ≤ 16, x ≥ 0 }
if the density at any point is proportional to its distance from the origin.
Total Charge
If an electric charge is distributed over a region D and the charge density (in units
of charge per unit area) is given by σ(x, y) at a point (x, y) in D, then the total charge
Q is given by
¨
σ(x, y) dA
Q=
D
Example 2. Charge is distributed over the part of the disk x2 + y 2 ≤ 1 in the rst
quadrant so that the charge density at (x, y) is σ(x, y) = x2 +y 2 measured in coulombs
per square meter (C/m2 ). Find the total charge.