1. Chapter 15 – Multiple Integrals 15.7 Triple Integrals 1 Objectives:  Understand how to calculate triple integrals  Understand and apply the use of triple integrals to different applications

2. Triple Integrals Just as we defined single integrals for functions of one variable and double integrals for functions of two variables, so we can define triple integrals for functions of three variables. 15.7 Triple Integrals2

3. Triple Integrals Let’s first deal with the simplest case where f is defined on a rectangular box: 15.7 Triple Integrals3

4. Triple Integrals The first step is to divide B into sub-boxes—by dividing: ◦ The interval [ a, b ] into l subintervals [ x i-1, x i ] of equal width Δx. ◦ [ c, d ] into m subintervals of width Δy. ◦ [ r, s ] into n subintervals of width Δz. 15.7 Triple Integrals4

5. Triple Integrals The planes through the endpoints of these subintervals parallel to the coordinate planes divide the box B into lmn sub-boxes ◦ Each sub-box has volume ΔV = Δx Δy Δz 15.7 Triple Integrals5

6. Triple Integrals Then, we form the triple Riemann sum where the sample point is in B ijk. 15.7 Triple Integrals6

7. Triple Integrals The triple integral of f over the box B is: if this limit exists. ◦ Again, the triple integral always exists if f is continuous. 15.7 Triple Integrals7

8. Fubini’s Theorem for Triple Integrals Just as for double integrals, the practical method for evaluating triple integrals is to express them as iterated integrals, as follows. If f is continuous on the rectangular box B = [a, b] x [c, d] x [r, s], then 15.7 Triple Integrals8

9. Fubini’s Theorem The iterated integral on the right side of Fubini’s Theorem means that we integrate in the following order: 1.With respect to x (keeping y and z fixed) 2.With respect to y (keeping z fixed) 3.With respect to z 15.7 Triple Integrals9

10. Example 1 – pg. 998 # 4 Evaluate the triple integral. 15.7 Triple Integrals10

11. Integral over a Bounded Region We restrict our attention to: ◦ Continuous functions f ◦ Certain simple types of regions 15.7 Triple Integrals11

12. Type I Region Eq.5 A solid region E is said to be of type 1 if it lies between the graphs of two continuous functions of x and y. That is, where D is the projection of E onto the xy -plane. 15.7 Triple Integrals12

13. Type I Region Notice that: ◦ The upper boundary of the solid E is the surface with equation z = u 2 (x, y). ◦ The lower boundary is the surface z = u 1 (x, y). 15.7 Triple Integrals13

14. Type I Region Eq. 6 If E is a type 1 region given by Equation 5, then we have Equation 6: 15.7 Triple Integrals14

15. Type II Region A solid region E is said to be of type 1 if it lies between the graphs of two continuous functions of x and y. That is, where D is the projection of E onto the yz -plane. 15.7 Triple Integrals15

16. Type II Region Notice that: ◦ The back surface is x = u 1 (y, z). ◦ The front surface is x = u 2 (y, z). 15.7 Triple Integrals16

17. Type II Region Eq. 10 For this type of region we have: 15.7 Triple Integrals17

18. Type III Region Finally, a type 3 region is of the form where: ◦ D is the projection of E onto the xz -plane. 15.7 Triple Integrals18

19. Type III Region Notice that: ◦ y = u 1 (x, z) is the left surface. ◦ y = u 2 (x, z) is the right surface. 15.7 Triple Integrals19

20. Type III Region Eq. 11 For this type of region, we have: 15.7 Triple Integrals20

21. Visualization Regions of Integration in Triple Integrals Regions of Integration in Triple Integrals 15.7 Triple Integrals21

22. Example 2 – pg. 998 # 10 Evaluate the triple integral. 15.7 Triple Integrals22

23. Example 3 – pg. 998 # 20 Use a triple integral to find the volume of the given solid. 15.7 Triple Integrals23

24. Example 4 – pg. 998 # 22 Use a triple integral to find the volume of the given solid. 15.7 Triple Integrals24

25. Example 5 – pg. 999 # 36 Write five other iterated integrals that are equal to the given iterated integral. 15.7 Triple Integrals25

26. More Examples The video examples below are from section 15.7 in your textbook. Please watch them on your own time for extra instruction. Each video is about 2 minutes in length. ◦ Example 1 Example 1 ◦ Example 3 Example 3 ◦ Example 6 Example 6 15.7 Triple Integrals26