Double integral - substitution help

[OrangeOne]

Calculate

?? (x/y) dxdy
D

where D is the area which is limited by the curves:
x^2+y^2=1
x^2+y^2=4
x= 0
x+y=0

When I draw this I get 4 different areas, the prof told me I could chose whichever I wanted.

Now, I think I should do variable substituion and say:

u = x^2+y^2 , u= 1 or 4
v= ??? How do I manage substitution for x= 0 and x+y=0?

When this is done I should use the formula:
?? ....dudv * J

Help please!

 

[galactus]

The region projected onto the xy-plane would look like the diagram below.

You could find the volume by using rectangular coordinates.

You could find the area of one section and then multiply by 2.

\(\displaystyle 2\int_{1}^{2}\int_{\sqrt{1-y^{2}}}^{\sqrt{4-y^{2}}}\frac{x}{y}dxdy\)

Or, in polar:

\(\displaystyle 4\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\int_{1}^{2}rcot({\theta})drd{\theta}\)