Question

Calculus w/ analytical geometry:

please be concise

- Fubini's Theorem: if a function is
continuous on the domain R, then the triple
integral can be evaluated in any order that
describes R.
(a) Explain the significance of this theorem.
(b) Provide an example to illustrate this
theorem.

- multi-integration
(a) Explain the purpose of changing variables
when double or triple integrating.
(b) Post an example illustrating such a change of variables.

Answer 1

Fubini's Theorem:

a) In some set of examples, we can do the integral in either direction. However, sometimes one direction of integration is significantly easier than the other so make sure that we think about which one you should do first before actually doing the integral.

b) Take example:

where

  

We can integrate into either direction for this example. But all integrals are not so easy to integrate in both direction.

- multi-integration
(a) Basically sometimes it is not to easy to find integration in either direction, in this situation we apply the change of variable method. That converts the given region into another region and in the converted region, the integration became much easier. In addition to converting the integrand into something simpler it will often also transform the region into one that is much easier to deal with.

b) Evaluate

  

where R is the trapezoidal region with vertices given by (0,0), (5,0), (5/2,5/2) and (5/2,−5/2) using the transformation

x = 2u + 3v and y = 2u − 3v.