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

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

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

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

Triple Integrals over General Bounded Regions The z-values are sandwiched between two functions: u 2 (x,y) and u 1 (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 General = “non parallelepiped” This is usually described by region “types” cleverly named type 1, type 2 or type 3!

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)

Example 3 Sketch, assign “type” and find the volume of the region created by the intersection of the cylinder x 2 + y 2 = 1 and the planes z = -1 and x + y + z = 4

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

The inner integral: = 5 - x - y = 5 

Example 4 (try this at home!) Evaluate in the region created by the intersection of the cylinder x 2 + y 2 = 1 and the planes z = -1 and x + y + z = 4 (answer is 0!)

Example 5 Find the volume bounded by the paraboloids y = x 2 + z 2 and y = 4 – x 2 - z 2

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