Working With Triple Integrals

1. Working With Triple Integrals Basics ideas – extension from 1D and 2D Iterated Integrals Extending to general bounded regions

2. Riemann Sums This is one way to define an iterated Integral over box B (what other ways can you think of?)

3. Example 1 Evaluate the following function over the region B = [0,3]×[-2,2] ×[1,3]

4. Example 2 What does mean if f(x,y,z) = 1

5. Triple Integrals over General Bounded Regions General = “non parallelepiped” • The z-values are sandwiched between two functions: u2(x,y) and u1(x,y) • This constrains z in the following way: • You can now use region D(x,y) to express x in terms of y or vice versa • Your final choice is determined by the range of one of the remaining variables This is usually described by region “types” cleverly named type 1, type 2 or type 3!

6. Region types • Type 1 if z is constrained between functions in (x,y) • Type 2 if x is constrained between functions in (y,z) • Type 3 if y is constrained between functions in (x,z)

7. Example 3 Sketch, assign “type” and find the volume of the region created by the intersection of the cylinder x2 + y2 = 1 and the planes z = -1 and x + y + z = 4

8. The iterated integral looks like this Note the pattern in the limits: “constant”  “function 1 variable”  “function in 2 variables”

9. The inner integral: = 5 - x - y = 5

10. Example 4 (try this at home!) Evaluate in the region created by the intersection of the cylinder x2 + y2 = 1 and the planes z = -1 and x + y + z = 4 (answer is 0!)

11. Example 5 Find the volume bounded by the paraboloids y = x2 + z2 and y = 4 – x2 - z2

12. y values are constrained by the two Paraboloids, so x & z values are constrained by the intersection of the paraboloids: