Planar Lamina

    A planar lamina is a flat plate that occupies a region in the plane that is so thin it can be regarded as two dimensional. If is the lamina's mass and is the area of  the region then is the density of the lamina (in units of mass per unit area). A lamina is homogeneous if its density is constant over and nonhomogeneous if varies from point to point.
    If is a continuous density function on the lamina corresponding to a plane region then the (total) mass of the planar lamina is given by



Physicists also consider other types of density that can be treated in the same manner. For example, if an electric charge is distributed over a region and the charge density (in units of charge per unit area) is given by at a point in then the total charge is given by  

Example (Planar Lamina) Charge is distributed over the triangular region described by and so that the charge density at is measured in coulombs per square meter. Find the total charge.
    
    Solution. We have



Thus the total charge is .

Example (Planar Lamina) Find the centroid of the planar lamina cut from the first quadrant by the line   
    
    Solution.
    

Example (Planar Lamina) The area of the region in the first quadrant bounded by the parabola and the line is 125/6 units. Find the centroid.
    
    Solution.
    

Example (Planar Lamina) Find the mass of the planar lamina occupying the smaller region cut from the ellipse by the parabola if
    
    Solution.
    

Example (Planar Lamina) Find the center of mass of the planar lamina bounded by the curves and if the density at the point is
    
    Solution.
    

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