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14_4 Calc III .docx

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Dragos Hrimiuc

Calculus III Section 14.4: Applications of Double Integrals Density and Mass Given a lamina (thin plate) occupies a region D in the xy-plane and its density (units of mass/unit area) at point (x, y) in D is given by ρ(x, y), where ρ is a continuous function on D, to find the total mass m: ∬ ( ) A similar application would be for electric charge over a region D. If the charge density (units of charge/unit area) is given by σ(x, y) at a point (x, y) in D, then the total charge Q is: ∬ ( ) Example 1: Charge is distributed over the triangular region D bounded by y = 1-x, x = 1, y = 1 and the char2e density at (x, y) is σ(x, y) = xy, measured in C/m . Find the total charge Moments and Center of Mass Recall: The moment of a particle about an axis is the product of its mass and its directed distance from the axis. So to get the moment of a lamina about the x-axis: ∬ ( ) Similarly, the moment of a lamina about the y-axis: ∬ ( ) The center of mass is the point (x*, y*) such that mx* = M ynd my* = M x In other words, the lamina behaves as if its entire mass is concentrated at its center of mass. Therefore: x* = (1/m)My= ∬ ( ) and y* = (1/m)Mx= ∬ ( ) Example 2: Find the mass and center of mass of a triangular lamina with vertices (0, 0), (1, 0) and (0, 2) if the density function is ρ(x, y) = 1 + 3x + y Example 3: The density at any point on a semicircular lamina is proportional to the distance from the center of the circle. Find the center of mass of the lamina. Moment of Inertia Also known as second moment Is the product of the mass of a particle and the square of its distance to the axis So to find the moment of inertia of a lamina about the x-axis or y-axis: ∬ ( ) ∬ ( ) We can also find the moment of inertia about the origin (called the polar moment of inertia) by: ∬( ) ( ) Example 4: Find the moments of inertia Ix, y 0 I of a homogeneous disk D with density ρ(x, y) = ρ, center at the origin, and radius a Probability We can now consider a pair of continuous random variables X and Y such as lifetimes of two components of a machine or the height/weight of randomly selected male students The joint density function of X and Y is a function f of two variables such that ( ) ∬ ( )
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