1. Chapter 12-Multiple Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

2. Chapter 12-Multiple Integrals 12.1 Double Integrals over Rectangular Regions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

3. Chapter 12-Multiple Integrals 12.1 Double Integrals over Rectangular Regions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved

4. Chapter 12-Multiple Integrals 12.1 Double Integrals over Rectangular Regions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved DEFINITION: Let f be a continuous function that is defined on the rectangle R = {(x, y) : a ≤x ≤ b, c ≤ y ≤ d}. For each positive integer N, let {x 0, x 1,..., x N } and {y 0, y 1,..., y N } be uniform partitions of the intervals [a, b] and [c, d] respectively. Set  x = (b − a) /N,  y = (d − c) /N, and  A = (  x) (  y). Let  ij be any point in subrectangle Q ij = [x i−1, x i ] × [y j−1, y j ]. Then the double integral of f over R is defined to be the limit of the Riemann sums as N tends to infinity.

5. Chapter 12-Multiple Integrals 12.1 Double Integrals over Rectangular Regions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Let f (x, y) = 6 − x 2 − y 2 for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 2. For each fixed x in the interval [0, 1],calculate For each fixed y in the interval [0, 2], calculate Iterated Integrals

6. Chapter 12-Multiple Integrals 12.1 Double Integrals over Rectangular Regions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved DEFINITION: The expressions And are called iterated integrals. Iterated Integrals EXAMPLES: Let f (x, y) = 6 − x 2 − y 2 for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 2. Calculate the iterated integrals and

7. Chapter 12-Multiple Integrals 12.1 Double Integrals over Rectangular Regions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Using Iterated Integrals to Calculate Double Integrals

8. Chapter 12-Multiple Integrals 12.1 Double Integrals over Rectangular Regions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Using Iterated Integrals to Calculate Double Integrals

9. Chapter 12-Multiple Integrals 12.1 Double Integrals over Rectangular Regions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz

10. Chapter 12-Multiple Integrals 12.2 Integration over More General Regions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Planar Regions Bounded by Finitely Many Curves

11. Chapter 12-Multiple Integrals 12.2 Integration over More General Regions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Planar Regions Bounded by Finitely Many Curves

12. Chapter 12-Multiple Integrals 12.2 Integration over More General Regions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Planar Regions Bounded by Finitely Many Curves

13. Chapter 12-Multiple Integrals 12.2 Integration over More General Regions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Changing the Order of Integration EXAMPLE: Calculate

14. Chapter 12-Multiple Integrals 12.2 Integration over More General Regions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Area of a Planar Region

15. Chapter 12-Multiple Integrals 12.2 Integration over More General Regions Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz

16. Chapter 12-Multiple Integrals 12.3 Calculation of Volumes of Solids Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved EXAMPLE: Calculate the volume below the plane 8x+4y +2z = 16, above the xy-plane, and in the first octant.

17. Chapter 12-Multiple Integrals 12.3 Calculation of Volumes of Solids Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Volume Between Two Surfaces

18. Chapter 12-Multiple Integrals 12.3 Calculation of Volumes of Solids Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved 1. Let V denote the volume of the solid that lies above the region R in the xy-plane, that is bounded above by z = 6 + x2 − y2, and that is bounded below by z = 1 + x2 + 2y. Then V is equal to for what function h (x, y)? 2. Calculate the volume of the solid that lies below the graph of f (x, y) = 2x + 3y 2 and over the rectangle [0, 1] × [0, 1] in the xy-plane. 3. Calculate the volume of the solid that lies below the graph of f (x, y) = 1+x, above the graph of g (x, y) = 2x, and over the rectangle [0, 1] × [0, 1] in the xy-plane. Quick Quiz

19. Chapter 12-Multiple Integrals 12.4 Polar Coordinates Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Point P has polar coordinates (2, 1) whereas point Q has Cartesian coordinates (2, 1)

20. Chapter 12-Multiple Integrals 12.4 Polar Coordinates Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Negative Values of the Radial Variable

21. Chapter 12-Multiple Integrals 12.4 Polar Coordinates Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Relating Polar Coordinates to Rectangular Coordinates EXAMPLE: Compute the rectangular coordinates of the point P whose polar coordinates are (4, 5  /6). EXAMPLE: Calculate all possible polar coordinates for the point Q with rectangular coordinates (−5, 5).

22. Chapter 12-Multiple Integrals 12.4 Polar Coordinates Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Graphing in Polar Coordinates EXAMPLE: Sketch the graph of r = . EXAMPLE: Sketch r = 3 (1 + cos (  )).

23. Chapter 12-Multiple Integrals 12.4 Polar Coordinates Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz

24. Chapter 12-Multiple Integrals 12.5 Integrating in Polar Coordinates Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Areas of More General Regions

25. Chapter 12-Multiple Integrals 12.5 Integrating in Polar Coordinates Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Areas of More General Regions EXAMPLE: Find the area inside the curve r = 2 sin (  ).

26. Chapter 12-Multiple Integrals 12.5 Integrating in Polar Coordinates Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Using Iterated Integrals to Calculate Area in Polar Coordinates

27. Chapter 12-Multiple Integrals 12.5 Integrating in Polar Coordinates Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Using Iterated Integrals to Calculate Area in Polar Coordinates

28. Chapter 12-Multiple Integrals 12.5 Integrating in Polar Coordinates Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Using Iterated Integrals to Calculate Area in Polar Coordinates EXAMPLE: Calculate the volume V of the solid bounded by the paraboloid z = 2x 2 + 2y 2 − 8 and the xy-plane.

29. Chapter 12-Multiple Integrals 12.5 Integrating in Polar Coordinates Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Change of Variable and the Jacobian

30. Chapter 12-Multiple Integrals 12.5 Integrating in Polar Coordinates Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Change of Variable and the Jacobian

31. Chapter 12-Multiple Integrals 12.5 Integrating in Polar Coordinates Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz

32. Chapter 12-Multiple Integrals 12.6 Triple Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Concept of the Triple Integral

33. Chapter 12-Multiple Integrals 12.6 Triple Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Concept of the Triple Integral

34. Chapter 12-Multiple Integrals 12.6 Triple Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved The Concept of the Triple Integral EXAMPLE: Integrate the function f(x, y, z) = 4x − 12z over the solid U = {(x, y, z) : 1 ≤ x ≤ 2, x ≤ y ≤ 2x, y − x ≤ z ≤ y}. DEFINITION: The volume of a solid U is defined to be when the integral exists.

35. Chapter 12-Multiple Integrals 12.6 Triple Integrals Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz

36. Chapter 12-Multiple Integrals 12.7 Physical Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Mass

37. Chapter 12-Multiple Integrals 12.7 Physical Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved First Moments

38. Chapter 12-Multiple Integrals 12.7 Physical Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Center of Mass

39. Chapter 12-Multiple Integrals 12.7 Physical Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Moment of Inertia EXAMPLE: Calculate the second moment I y =0 of the lamina with constant density  = 3 that occupies the region bounded by the curves y = −x 2 + 4 and y = 0.

40. Chapter 12-Multiple Integrals 12.7 Physical Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Mass, First Moment, Moment of Inertia, and Center of Mass in Three Dimensions

41. Chapter 12-Multiple Integrals 12.7 Physical Applications Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz

42. Chapter 12-Multiple Integrals 12.8 Other Coordinate Systems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Cylindrical Coordinates EXAMPLE: The point P has rectangular coordinates (1, 1, 4). What are its cylindrical coordinates?

43. Chapter 12-Multiple Integrals 12.8 Other Coordinate Systems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Spherical Coordinates

44. Chapter 12-Multiple Integrals 12.8 Other Coordinate Systems Calculus, 2ed, by Blank & Krantz, Copyright 2011 by John Wiley & Sons, Inc, All Rights Reserved Quick Quiz