What Are the Steps for Using Change of Variables to Find Bounded Regions?

[mattbonner]

Homework Statement

using change of variables, find the region bounded by
y = x, y = 2x, xy = 1, xy=2

Homework Equations

The Attempt at a Solution

i know i have to introduce the variables u, v
the problem is i don't understand how to introduce them
i tried read the textbook but the examples all introduce the new variables for you

i think i understand the steps that follow, I am just stuck at the beginning

 

[LCKurtz]

Well, you would like your uv integral to have constant limits if possible. You have xy=1 and xy=2 so that certainly suggests setting u or v = xy. If you use v, then v will go from 1 to 2.

Now can you see how to choose u so that y = x and y = 2x give you constant limits for u?

 

[mattbonner]

i guess y/x?
that'll give me 1 to 2 as well?

 

[LCKurtz]

Good guess. I hope it wasn't a pure guess. You don't have to ask about the limits. Look at the equations. And plot your uv region.

 

[mattbonner]

i vaguely remembered a similar example the prof did, i missed how he had chosen u and v, which resulted in my confusion

thank you so much for the help!