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Working With Triple Integrals

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.

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Working With Triple Integrals

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  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:

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