Physical Applications of Double Integrals

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Mass and Static Moments of a Lamina

Suppose we have a lamina which occupies a region \(R\) in the \(xy\)-plane and is made of non-homogeneous material. Its density at a point \(\left( {x,y} \right)\) in the region \(R\) is \(\rho \left( {x,y} \right).\) The total mass of the lamina is expressed through the double integral as follows:

\[m = \iint\limits_R {\rho \left( {x,y} \right)dA} .\]

The static moment of the lamina about the \(x\)-axis is given by the formula

\[{M_x} = \iint\limits_R {y\rho \left( {x,y} \right)dA} .\]

Similarly, the static moment of the lamina about the \(y\)-axis is

\[{M_y} = \iint\limits_R {x\rho \left( {x,y} \right)dA} .\]

The coordinates of the center of mass of a lamina occupying the region \(R\) in the \(xy\)-plane with density function \(\rho \left( {x,y} \right)\) are described by the formulas

\[\bar x = \frac{{{M_y}}}{m} = \frac{1}{m}\iint\limits_R {x\rho \left( {x,y} \right)dA} = \frac{{\iint\limits_R {x\rho \left( {x,y} \right)dA} }}{{\iint\limits_R {\rho \left( {x,y} \right)dA} }},\]

\[\bar y = \frac{{{M_x}}}{m} = \frac{1}{m}\iint\limits_R {y\rho \left( {x,y} \right)dA} = \frac{{\iint\limits_R {y\rho \left( {x,y} \right)dA} }}{{\iint\limits_R {\rho \left( {x,y} \right)dA} }}.\]

When the mass density of the lamina is \(\rho \left( {x,y} \right) = 1\) for all \(\left( {x,y} \right)\) in the region \(R,\) the center of mass is defined only by the shape of the region and is called the centroid of \(R.\)

Moments of Inertia of a Lamina

The moment of inertia of a lamina about the \(x\)-axis is defined by the formula

\[{I_x} = \iint\limits_R {{y^2}\rho \left( {x,y} \right)dA} .\]

Similarly, the moment of inertia of a lamina about the \(y\)-axis is given by

\[{I_y} = \iint\limits_R {{x^2}\rho \left( {x,y} \right)dA} .\]

The polar moment of inertia is

\[{I_0} = \iint\limits_R {\left( {{x^2} + {y^2}} \right)\rho \left( {x,y} \right)dA} .\]

Charge of a Plate

Suppose electrical charge is distributed over a region which has area \(R\) in the \(xy\)-plane and its charge density is defined by the function \({\sigma \left( {x,y} \right)}.\) Then the total charge \(Q\) of the plate is defined by the expression

\[Q = \iint\limits_R {\sigma \left( {x,y} \right)dA} .\]

Average Value of a Function

We give here the formula for calculation of the average value of a distributed function. Let \({f \left( {x,y} \right)}\) be a continuous function over a closed region \(R\) in the \(xy\)-plane. The average value \(\mu\) of the function \({f \left( {x,y} \right)}\) in the region \(R\) is given by the formula

\[\mu = \frac{1}{S}\iint\limits_R {f\left( {x,y} \right)dA} ,\]

where \(S = \iint\limits_R {dA} \) is the area of the region of integration \(R.\)