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Chapter 15 – Multiple Integrals 15.5 Applications of Double Integrals Objectives: Understand the physical applications of double integrals Dr. Erickson 15.5 Applications of Double Integrals 1 Applications of Double Integrals In this section, we explore physical applications—such as computing: ◦ Mass ◦ Electric charge ◦ Center of mass ◦ Moment of inertia Dr. Erickson 15.5 Applications of Double Integrals 2 Density and Mass In Section 8.3, we used single integrals to compute moments and the center of mass of a thin plate or lamina with constant density. Now, equipped with the double integral, we can consider a lamina with variable density. 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. Dr. Erickson 15.5 Applications of Double Integrals 3 Mass This means that: ( x , y ) lim m A 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. Dr. Erickson 15.5 Applications of Double Integrals 4 Mass If we now increase the number of subrectangles, we obtain the total mass m of the lamina as the limiting value of the approximations: k m lim k ,l l i 1 ( x ij , y ij ) A * * j 1 ( x , y ) dA D Dr. Erickson 15.5 Applications of Double Integrals 5 Density and Mass Physicists also consider other types of density that can be treated in the same manner. For example, an electric charge is distributed over a region D and the charge density (in units of charge per unit area) is given by σ(x, y) at a point (x, y) in D. Dr. Erickson 15.5 Applications of Double Integrals 6 Total Charge Then, the total charge Q is given by: Q ( x , y ) d A D Dr. Erickson 15.5 Applications of Double Integrals 7 Example 1 Electric charge is distributed over the disk x2 + y2 4 so that the charge density at (x, y) is (x, y) is (x, y) = x + y + x2 + y2 (measured in coulombs per square meter). Find the total charge on the disk. Dr. Erickson 15.5 Applications of Double Integrals 8 Moments and Centers of Mass In Section 8.3, we found the center of mass of a lamina with constant density. Here, we consider a lamina with variable density. Suppose the lamina occupies a region D and has density function ρ(x, y). ◦ Recall from Chapter 8 that we defined the moment of a particle about an axis as the product of its mass and its directed distance from the axis. Dr. Erickson 15.5 Applications of Double Integrals 9 Moments and Center of Mass We divide D into small rectangles as earlier. Then, the mass of Rij is approximately: ρ(xij*, yij*) ∆A So, we can approximate the moment of Rij with respect to the x-axis by: [ρ(xij*, yij*) ∆A] yij* Dr. Erickson 15.5 Applications of Double Integrals 10 Moment about the x-axis If we now add these quantities and take the limit as the number of sub rectangles becomes large, we obtain the moment of the entire lamina about the x-axis: m M x lim m,n n i 1 y ( x , y ) A * ij * ij * ij j 1 y ( x , y ) dA D Dr. Erickson 15.5 Applications of Double Integrals 11 Moment about the y-axis Similarly, the moment about the y-axis is: m M y lim m ,n n x i 1 * ij ( x , y ) A * ij * ij j 1 x ( x , y ) dA D Dr. Erickson 15.5 Applications of Double Integrals 12 Center of Mass As before, we define the center of mass ( x , y ) so that m x M y and m y M x . The physical significance is that the lamina behaves as if its entire mass is concentrated at its center of mass. Thus, the lamina balances horizontally when supported at its center of mass. Dr. Erickson 15.5 Applications of Double Integrals 13 Center of Mass x The coordinates ( x , y ) of the center of mass of a lamina occupying the region D and having density function ρ(x, y) are: M y m 1 x ( x , y ) dA m y M m D x 1 y ( x , y ) dA m D where the mass m is given by: m ( x , y ) d A D Dr. Erickson 15.5 Applications of Double Integrals 14 Example 2 Find the mass and center of mass of the lamina that occupies the region D and has the given density function . D is the triangular region enclosed by th e lines x 0, y x , and 2 x y 6; x , y x Dr. Erickson 2 15.5 Applications of Double Integrals 15 Moment of Inertia The moment of inertia (also called the second moment) of a particle of mass m about an axis is defined to be mr2, where r is the distance from the particle to the axis. ◦ We extend this concept to a lamina with density function ρ(x, y) and occupying a region D by proceeding as we did for ordinary moments. Dr. Erickson 15.5 Applications of Double Integrals 16 Moment of Inertia (x-axis) The result is the moment of inertia of the lamina about the x-axis: m I x lim m,n y 2 n (y i 1 * ij ) (x , y ) A 2 * ij * ij j 1 ( x , y ) dA D Dr. Erickson 15.5 Applications of Double Integrals 17 Moment of Inertia (y-axis) Similarly, the moment of inertia about the y-axis is given by: m I y lim m ,n x 2 n i 1 ( x ij ) ( x ij , y ij ) A * 2 * * j 1 ( x , y ) dA D Dr. Erickson 15.5 Applications of Double Integrals 18 Moment of Inertia (Origin) It is also of interest to consider the moment of inertia about the origin (also called the polar moment of inertia): m I 0 lim m ,n ( x 2 n [( x i 1 * ij ) ( y ) ] ( x , y ) A 2 * ij 2 * ij * ij j 1 y ) ( x , y ) dA 2 D ◦ Note that I0 = Ix + Iy. Dr. Erickson 15.5 Applications of Double Integrals 19 Example 3 Find the moments of inertia Ix , Iy , Io for the lamina of the problem below. D is bounded by e , y 0, x 0, and x 1; x x, y y Dr. Erickson 15.5 Applications of Double Integrals 20 Example 4 A lamina occupies the part of the disk x2 + y2 1 in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis. Dr. Erickson 15.5 Applications of Double Integrals 21 Radius of Gyration Dr. Erickson 15.5 Applications of Double Integrals 22 Radius of Gyration Dr. Erickson 15.5 Applications of Double Integrals 23 Example 5 A lamina with constant density (x, y) = occupies the given region. Find the moments of inertia Ix , Iy , Io and the radii of gyration. The part of the disk x2 + y2 ≤ a2 in the first quadrant. Dr. Erickson 15.5 Applications of Double Integrals 24 Probability and Expected Values Please read through these pages in your text book. Dr. Erickson 15.5 Applications of Double Integrals 25 Dr. Erickson 15.5 Applications of Double Integrals 26 More Examples The video examples below are from section 15.5 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦ Example 2 ◦ Example 3 ◦ Example 4 Dr. Erickson 15.5 Applications of Double Integrals 27 Demonstrations Feel free to explore these demonstrations below. ◦ Center of Mass of a Polygon ◦ Moment of Inertia Dr. Erickson 15.5 Applications of Double Integrals 28