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Section 13.6: Applications of Double Integrals In this section, we explore physical applications of double integrals such as computing mass, center of mass, and electrical charge. Theorem: (Total Electrical Charge) Suppose an electrical charge is distributed over a region D in the xy-plane and the charge density at a point (x, y) ∈ D is given by σ(x, y). Then the total charge is given by ZZ Q= σ(x, y)dA. D Example: Electric charge is distributed over the unit disk x2 + y 2 ≤ 1 so that the charge density at (x, y) is σ(x, y) = 1 + x2 + y 2 (measured in coulombs per square meter). Find the total charge on the disk. Using polar coordinates, the total charge is ZZ Q = σ(x, y)dA D Z 2π Z 1 = (1 + r2 )rdrdθ 0 Z 10 = 2π (r + r3 )dr 0 1 1 2 1 4 r + r = 2π 2 4 0 3π = C. 2 Theorem: (Center of Mass of a Lamina) Suppose a lamina occupies some region D in the xy-plane and its density at a point (x, y) ∈ D is given by ρ(x, y). Then the total mass of the lamina is given by ZZ m= ρ(x, y)dA. D The coordinates (x̄, ȳ) of the center of mass of the lamina are ZZ ZZ 1 1 x̄ = xρ(x, y)dA ȳ = yρ(x, y)dA. m D m D Example: √ Find the mass and center of mass of the lamina that occupies the region D bounded by y = x, y = 0, and x = 1 with density function ρ(x, y) = 3x. The mass of the lamina is ZZ ρ(x, y)dA m = Z D √ 1Z x 3xdydx = 0 Z = = 0 1 3x3/2 dx 0 1 6 5/2 x 5 0 6 = . 5 Using the formulas in the previous theorem, ZZ 1 x̄ = xρ(x, y)dA m D Z Z √x 5 1 = 3x2 dydx 6 0 0 Z 5 1 5/2 = x dx 2 0 1 5 7/2 = x 7 0 5 = 7 and ZZ 1 yρ(x, y)dA ȳ = m D Z Z √x 5 1 = 3xydydx 6 0 0 √x Z 5 1 2 xy dx = 4 0 0 Z 1 5 x2 dx = 4 0 1 5 2 = x 12 0 5 = . 12 The mass of the lamina is m = 6 5 and the center of mass is (x̄, ȳ) = 5 5 , 7 12 .