Multiple Integration 14 Copyright © Cengage Learning. All rights reserved.

Triple Integrals The first part of the slide show is to introduce you to the material in Section 14.6 At the end of the slide show you will be told how to complete Quiz #5 Copyright © Cengage Learning. All rights reserved. 14.6

3  Complete this work early such as in the time period you would have been in class on Tuesday, November 23, then enjoy your holiday.  HAPPY THANKSGIVING!

4 Use a triple integral to find the volume of a solid region. Objectives

5 Triple Integrals

6 The procedure used to define a triple integral follows that used for double integrals. Consider a function f of three variables that is continuous over a bounded solid region Q. Then, encompass Q with a network of boxes and form the inner partition consisting of all boxes lying entirely within Q, as shown in Figure Figure Triple Integrals

7 The volume of the ith box is The norm of the partition is the length of the longest diagonal of the n boxes in the partition. Choose a point (x i, y i, z i ) in each box and form the Riemann sum Triple Integrals

8 Taking the limit as leads to the following definition. Triple Integrals

9 Some of the properties of double integrals can be restated in terms of triple integrals. Triple Integrals

10 Triple Integrals

11 Example 1 – Evaluating a Triple Iterated Integral Evaluate the triple iterated integral Solution: For the first integration, hold x and y constant and integrate with respect to z.

12 Example 1 – Solution For the second integration, hold x constant and integrate with respect to y. Finally, integrate with respect to x. cont’d

13 To find the limits for a particular order of integration, it is generally advisable first to determine the innermost limits, which may be functions of the outer two variables. Then, by projecting the solid Q onto the coordinate plane of the outer two variables, you can determine their limits of integration by the methods used for double integrals. Triple Integrals

14 For instance, to evaluate first determine the limits for z, and then the integral has the form Triple Integrals

15 By projecting the solid Q onto the xy-plane, you can determine the limits for x and y as you did for double integrals, as shown in Figure Figure Triple Integrals

16 Go to  Click Enter

17 Go to Choose a Book Drop Down Menu

18 Choose Varberg: Calculus, 9e Click Submit

19 Choose Chapter 13 Section 7

20 Click on Exercise 1

21 Try Problems 1,3,4,7 listed at the top of the screen

22 Additional Help  Use the Help Me Solve This and View an Example buttons at the side for additional assistance This will supplement the homework from Section 13.6

23 QUIZ 5  Now go back to the chapter list in InteractMath and choose Chapter 13 Section 3.  You will need to complete Exercises 3, 5, 16, and 19 listed in the Exercise list. –Note: Once you choose Exercise 3 and are in the problem list you will need to work on #1-4, these correspond to the exercises listed above. –These are like problems you did from homework in Sections 14.1 & 14.2  Do your work for each problem neatly on a piece of notebook paper

24 QUIZ 5  Use the Help Me Solve This and View an Example buttons at the side for additional assistance  Once you have completed all 4 problems hit the Close Button at the bottom of the screen.  This will return you to the list of Exercises and there should be check marks next to the problems you completed correctly. –If you do not complete the quiz in one sitting be sure to print the summary page each time before you leave InteractMath as the check marks will disappear when you close the program  Print this page and staple it to the top of your notebook paper containing the work for each problem –Note: There is a total of 4 problems on this quiz  Bring it to class Tuesday, November 30 as this is Quiz 5

25 Print pages like this showing that you have completed Exercises 3, 5, 16, & 19