1. Chapter 15 – Multiple Integrals 15.5 Applications of Double Integrals • Objectives: • Understand the physical applications of double integrals 15.5 Applications of Double Integrals

2. Applications of Double Integrals • In this section, we explore physical applications—such as computing: • Mass • Electric charge • Center of mass • Moment of inertia 15.5 Applications of Double Integrals

3. 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 Dof 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 ρis a continuous function on D. 15.5 Applications of Double Integrals

4. Mass • This means that: • where: • Δmand ΔAare the mass and area of a small rectangle that contains (x, y). • The limit is taken as the dimensions of the rectangle approach 0. 15.5 Applications of Double Integrals

5. 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: 15.5 Applications of Double Integrals

6. 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. 15.5 Applications of Double Integrals

7. Total Charge • Then, the total charge Q is given by: 15.5 Applications of Double Integrals

8. Example 1 – pg. 1012 # 2 • 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. 15.5 Applications of Double Integrals

9. 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. 15.5 Applications of Double Integrals

10. Moments and Center of Mass • We divide D into small rectangles as earlier. • Then, the mass ofRijis 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* 15.5 Applications of Double Integrals

11. 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 momentof the entire lamina about the x-axis: 15.5 Applications of Double Integrals

12. Moment about the y-axis • Similarly, the moment about the y-axis is: 15.5 Applications of Double Integrals

13. Center of Mass • As before, we define the center of mass so that and . • 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. 15.5 Applications of Double Integrals

14. Center of Mass • The coordinates of the center of mass of a lamina occupying the region Dand having density function ρ(x, y) are: where the mass m is given by: 15.5 Applications of Double Integrals

15. Example 2 – pg. 1012 # 6 • Find the mass and center of mass of the lamina that occupies the region D and has the given density function . 15.5 Applications of Double Integrals

16. 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. 15.5 Applications of Double Integrals

17. Moment of Inertia (x-axis) • The result is the moment of inertiaof the lamina about the x-axis: 15.5 Applications of Double Integrals

18. Moment of Inertia (y-axis) • Similarly, the moment of inertia about the y-axisis given by: 15.5 Applications of Double Integrals

19. 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): • Note that I0 = Ix + Iy. 15.5 Applications of Double Integrals

20. Example 3 • Find the moments of inertia Ix , Iy, Iofor the lamina of exercise 7. 15.5 Applications of Double Integrals

21. Example 4 – pg. 1012 #11 • 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. 15.5 Applications of Double Integrals

22. 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 15.5 Applications of Double Integrals

23. Demonstrations • Feel free to explore these demonstrations below. • Center of Mass of a Polygon • Moment of Inertia 15.5 Applications of Double Integrals