2. CHAPTER 13 Multiple Integrals Slide 2 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 13.1DOUBLE INTEGRALS 13.2AREA, VOLUME AND CENTER OF MASS 13.3DOUBLE INTEGRALS IN POLAR COORDINATES 13.4SURFACE AREA 13.5TRIPLE INTEGRALS 13.6CYLINDRICAL COORDINATES 13.7SPHERICAL COORDINATES 13.8CHANGE OF VARIABLES IN MULTIPLE INTEGRALS

3. 13.2AREA, VOLUME AND CENTER OF MASS Area as a Double Integral Slide 3 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

4. EXAMPLE 13.2AREA, VOLUME AND CENTER OF MASS 2.1Using a Double Integral to Find Area Slide 4 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Find the area of the plane region bounded by the graphs of x = y 2, y − x = 3, y = −3 and y = 2.

5. EXAMPLE Solution 13.2AREA, VOLUME AND CENTER OF MASS 2.1Using a Double Integral to Find Area Slide 5 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

6. EXAMPLE 13.2AREA, VOLUME AND CENTER OF MASS 2.2Using a Double Integral to Find Volume Slide 6 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Find the volume of the tetrahedron bounded by the plane 2x + y + z = 2 and the three coordinate planes.

7. EXAMPLE Solution 13.2AREA, VOLUME AND CENTER OF MASS 2.2Using a Double Integral to Find Volume Slide 7 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

8. EXAMPLE 13.2AREA, VOLUME AND CENTER OF MASS 2.3Finding the Volume of a Solid Slide 8 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Find the volume of the solid lying in the first octant and bounded by the graphs of z = 4 − x 2, x + y = 2, x = 0, y = 0 and z = 0.

9. EXAMPLE Solution 13.2AREA, VOLUME AND CENTER OF MASS 2.3Finding the Volume of a Solid Slide 9 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

10. EXAMPLE 13.2AREA, VOLUME AND CENTER OF MASS 2.4Finding the Volume of a Solid Bounded Above the xy-Plane Slide 10 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Find the volume of the solid bounded by the graphs of z = 2, z = x 2 + 1, y = 0 and x + y = 2.

11. EXAMPLE Solution 13.2AREA, VOLUME AND CENTER OF MASS 2.4Finding the Volume of a Solid Bounded Above the xy-Plane Slide 11 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

12. EXAMPLE 13.2AREA, VOLUME AND CENTER OF MASS 2.5Estimating Population Slide 12 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Suppose that models the population density (population per square mile) of a species of small animals, with x and y measured in miles. Estimate the population in the triangular-shaped habitat with vertices (1, 1), (2, 1) and (1, 0).

13. EXAMPLE Solution 13.2AREA, VOLUME AND CENTER OF MASS 2.5Estimating Population Slide 13 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

14. 13.2AREA, VOLUME AND CENTER OF MASS Moments and Center of Mass Slide 14 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Consider a thin, flat plate (a lamina) in the shape of the region R ⊂ whose density (mass per unit area) varies throughout the plate (i.e., some areas of the plate are more dense than others). From an engineering standpoint, it’s often important to determine where you could place a support to balance the plate. We call this point the center of mass of the lamina. We’ll first need to find the total mass of the plate.

15. 13.2AREA, VOLUME AND CENTER OF MASS Moments and Center of Mass Slide 15 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

16. 13.2AREA, VOLUME AND CENTER OF MASS Moments and Center of Mass Slide 16 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

17. EXAMPLE 13.2AREA, VOLUME AND CENTER OF MASS 2.6Finding the Center of Mass of a Lamina Slide 17 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Find the center of mass of the lamina in the shape of the region bounded by the graphs of y = x 2 and y = 4, having mass density given by ρ(x, y) = 1 + 2y + 6x 2.

18. EXAMPLE Solution 13.2AREA, VOLUME AND CENTER OF MASS 2.6Finding the Center of Mass of a Lamina Slide 18 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

19. EXAMPLE Solution 13.2AREA, VOLUME AND CENTER OF MASS 2.6Finding the Center of Mass of a Lamina Slide 19 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

20. EXAMPLE Solution 13.2AREA, VOLUME AND CENTER OF MASS 2.6Finding the Center of Mass of a Lamina Slide 20 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

21. EXAMPLE Solution 13.2AREA, VOLUME AND CENTER OF MASS 2.6Finding the Center of Mass of a Lamina Slide 21 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

22. 13.2AREA, VOLUME AND CENTER OF MASS Second Moments Slide 22 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. moment of inertia about the x-axis moment of inertia about the y-axis Physics tells us that the larger I y is, the more difficult it is to rotate the lamina about the y-axis. Similarly, the larger I x is, the more difficult it is to rotate the lamina about the x-axis.

23. EXAMPLE 13.2AREA, VOLUME AND CENTER OF MASS 2.7Finding the Moments of Inertia of a Lamina Slide 23 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Find the moments of inertia I y and I x for the lamina in example 2.6.

24. EXAMPLE Solution 13.2AREA, VOLUME AND CENTER OF MASS 2.7Finding the Moments of Inertia of a Lamina Slide 24 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

25. EXAMPLE Solution 13.2AREA, VOLUME AND CENTER OF MASS 2.7Finding the Moments of Inertia of a Lamina Slide 25 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

26. EXAMPLE Solution 13.2AREA, VOLUME AND CENTER OF MASS 2.7Finding the Moments of Inertia of a Lamina Slide 26 © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. It is much more difficult to rotate the lamina about the x- axis than about the y-axis.