1. ESSENTIAL CALCULUS CH12 Multiple integrals

2. In this Chapter: 12.1 Double Integrals over Rectangles 12.2 Double Integrals over General Regions 12.3 Double Integrals in Polar Coordinates 12.4 Applications of Double Integrals 12.5 Triple Integrals 12.6 Triple Integrals in Cylindrical Coordinates 12.7 Triple Integrals in Spherical Coordinates 12.8 Change of Variables in Multiple Integrals Review

3. Chapter 12, 12.1, P665 (See Figure 2.) Our goal is to find the volume of S.

4. Chapter 12, 12.1, P667

6. 5. DEFINITION The double integral of f over the rectangle R is if this limit exists.

7. Chapter 12, 12.1, P668

8. If f (x, y) ≥ 0, then the volume V of the solid that lies above the rectangle R and below the surface z=f (x, y) is

9. Chapter 12, 12.1, P669 MIDPOINT RULE FOR DOUBLE INTEGRALS where x i is the midpoint of [x i-1, x i ] and y j is the midpoint of [y j-1, y j ].

10. Chapter 12, 12.1, P672 10. FUBINI ’ S THEOREM If f is continuous on the rectangle R={(x, y} │ a ≤ x≤ b, c ≤ y ≤d}, then More generally, this is true if we assume that f is bounded on R, is discontinuous only on a finite number of smooth curves, and the iterated integrals exist.

11. Chapter 12, 12.1, P673 where R

12. Chapter 12, 12.2, P676

14. Chapter 12, 12.2, P677

16. 3.If f is continuous on a type I region D such that then

17. Chapter 12, 12.2, P678

18. where D is a type II region given by Equation 4.

19. Chapter 12, 12.2, P681

20. 6. 7. If f (x, y) ≥ g (x, y) for all (x, y) in D, then 8. If D=D 1 U D 2, where D 1 and D 2 don ’ t overlap except perhaps on their boundaries (see Figure 17), then

21. Chapter 12, 12.2, P682

22. 11. If m≤ f (x, y) ≤ M for all (x,y) in D, then

23. Chapter 12, 12.3, P684

26. r 2 =x 2 +y 2 x=r cosθ y=r sinθ

27. Chapter 12, 12.3, P685

28. Chapter 12, 12.3, P686 2. CHANGE TO POLAR COORDINATES IN A DOUBLE INTEGRAL If f is continuous on a polar rectangle R given by 0≤a≤r≤b, a≤θ≤ β, where 0≤ β-α ≤2, then

29. Chapter 12, 12.3, P687 3. If f is continuous on a polar region of the form then

30. Chapter 12, 12.5, P696 3. DEFINITION The triple integral of f over the box B is if this limit exists.

31. Chapter 12, 12.5, P696

32. 4. FUBINI ’ S THEOREM FOR TRIPLE INTEGRALS If f is continuous on the rectangular box B=[a, b] ╳ [c, d ] ╳ [r, s], then

33. Chapter 12, 12.5, P697

35. Chapter 12, 12.5, P698

37. Chapter 12, 12.5, P699

38. Chapter 12, 12.5, P700

39. Chapter 12, 12.6, P705

40. To convert from cylindrical to rectangular coordinates, we use the equations 1 x=r cosθ y=r sinθ z=z whereas to convert from rectangular to cylindrical coordinates, we use 2. r 2 =x 2 +y 2 tan θ= z=z

41. Chapter 12, 12.6, P706

44. Chapter 12, 12.6, P707 formula for triple integration in cylindrical coordinates.

45. Chapter 12, 12.7, P709

49. Chapter 12, 12.7, P710

52. Chapter 12, 12.7, P711 FIGURE 8 Volume element in spherical coordinates: dV=p 2 sinødpd Θd ø

53. Chapter 12, 12.7, P711 where E is a spherical wedge given by Formula for triple integration in spherical coordinates

54. Chapter 12, 12.8, P716

55. Chapter 12, 12.8, P717

56. Chapter 12, 12.8, P718 7. DEFINITION The Jacobian of the transformation T given by x= g (u, v) and y= h (u, v) is

57. Chapter 12, 12.8, P719 9. CHANGE OF VARIABLES IN A DOUBLE INTEGRAL Suppose that T is a C 1 transformation whose Jacobian is nonzero and that maps a region S in the uv-plane onto a region R in the xy-plane. Suppose that f is continuous on R and that R and S are type I or type II plane regions. Suppose also that T is one-to-one, except perhaps on the boundary of. S. Then

58. Chapter 12, 12.8, P721 Let T be a transformation that maps a region S in uvw-space onto a region R in xyz-space by means of the equations x=g (u, v, w) y=h (u, v, w) z=k (u, v, w) The Jacobian of T is the following 3X3 determinant: 12.

59. Chapter 12, 12.8, P722 13.