Wacky change of variables for Multi integration

[Theelectricchild]

Wacky change of variables for Multi integration!

Arghh I am having diffiiculty with these problems.
I am having difficulty mastering the LaTeX form--- (things like how to make a double integral etc) so

if you look at this site

http://www.math.washington.edu/~m124/Stewart5Eprobs/5ET-15problems.pdf

and look at page 1040 in the text (ch 15.9) the problem in question is number 14...

I am having a really tough time rewriting the integrand in u,v form with the given transformation. Perhaps there was a mistake in my algebra--- but what would be the way to go about doing this?

and also for number 20... would the proper transformation be u = x+y and v = x^2 - y^2 ? Or would something else work better.

Again I apologize for posting a link--- I promise I will take the time to learn LaTeX before I post--- I just have a tough time with Change Of Variables overall.

Thanks for all your help.

 

[Theelectricchild]

Oops, make that Page 1048, not 1040---

 

[Theelectricchild]

haha nm i figured out 20--- its better to expand the x^2 - y^2 ... makes life good.

 

[cookiemonster]

For #14, I get the intermediate integral

$$2\int_R(u^2 + v^2) \frac{\partial (x,y)}{\partial (u,v)} \, dA$$
where
$$R: \quad u^2 + v^2 = 1$$

Which can then be changed into polar coordinates to be evaluated.

For #20, I'd try the substitutions u = x + y and v = x - y.

cookiemonster

 

[Theelectricchild]

Wow thanks cookie I got it! The integrand turns out nicely coz the region is a simple circle--- and easily evaluated using polor coordinates!