Da Nang-2/2015 Natural Science Department – Duy Tan University Lecturer: Ho Xuan Binh Applications of Double Integrals In this section, we will learn about: The physical applications of double integrals.

Natural Science Department – Duy Tan University  In this section, we explore physical applications—such as computing:  Mass  Center of mass

DENSITY 1 Natural Science Department – Duy Tan University  Suppose the lamina occupies a region D of the xy-plane.  Also, let its density (in units of mass per unit area) at a point (x, y) in D be given by ρ(x, y), where ρ is a continuous function on D. Applications of Double integrals

MASS 2 Natural Science Department – Duy Tan University  This means that: where:  Δm and ΔA are the mass and area of a small rectangle that contains (x, y).  The limit is taken as the dimensions of the rectangle approach 0. Applications of Double integrals

MASS 2 Natural Science Department – Duy Tan University  To find the total mass m of the lamina, we:  Divide a rectangle R containing D into subrectangles R ij of equal size.  Consider ρ(x, y) to be 0 outside D. Applications of Double integrals

MASS 2 Natural Science Department – Duy Tan University  If we choose a point (x ij *, y ij *) in R ij, then the mass of the part of the lamina that occupies R ij is approximately ρ(x ij *, y ij *) ∆A where ∆A is the area of R ij. Applications of Double integrals

MASS 2 Natural Science Department – Duy Tan University  If we add all such masses, we get an approximation to the total mass: Applications of Double integrals

MASS 2 Natural Science Department – Duy Tan University  If we now increase the number of subrectangles, we obtain the total mass m of the lamina as the limiting value of the approximations: Applications of Double integrals

MOMENTS AND CENTERS OF MASS 3 Natural Science Department – Duy Tan University  We divide D into small rectangles as earlier.  Then, the mass of R ij is approximately: ρ(x ij *, y ij *) ∆A  So, we can approximate the moment of R ij with respect to the x-axis by: [ρ(x ij *, y ij *) ∆A] y ij * Applications of Double integrals

MOMENT ABOUT X-AXIS 4 Natural Science Department – Duy Tan University If we now add these quantities and take the limit as the number of subrectangles becomes large, we obtain the moment of the entire lamina about the x-axis: Applications of Double integrals

MOMENT ABOUT Y-AXIS 5 Natural Science Department – Duy Tan University Similarly, the moment about the y-axis is: Applications of Double integrals

CENTER OF MASS 6 Natural Science Department – Duy Tan University As before, we define the center of mass so that and. Applications of Double integrals

CENTER OF MASS 6 Natural Science Department – Duy Tan University  The coordinates of the center of mass of a lamina occupying the region D and having density function ρ(x, y) are: where the mass m is given by: Applications of Double integrals

Example 7 Natural Science Department – Duy Tan University Applications of Double integrals  Find the mass and center of mass of a triangular lamina with vertices (0, 0), (1, 0), (0, 2) and if the density function is ρ(x, y) = 1 + 3x + y

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