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NewtonianAlch
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Homework Statement
Find the area in the positive quadrant of the x-y plane bounded by the curves [itex]{x}^{2}+2\,{y}^{2}=1[/itex], [itex]{x}^{2}+2\,{y}^{2}=4[/itex], [itex]y=2\,x[/itex], [itex]y=5\,x[/itex]
The Attempt at a Solution
This is a graph of the region:
http://img21.imageshack.us/img21/2947/59763898.jpg
One thing I was confused about in regards to the domain was which part to specifically shade in, since there are two ellipses here. I can see that the relevant area is in between the two lines, but now there are two boundaries due to the ellipses. I'm guessing it's in the inner one, and not both, because the smaller ellipse doesn't contain anything outside it.
For the substitutions I thought of [itex]u = {x}^{2}+2\,{y}^{2}[/itex] and [itex] v = {\frac {y}{x}}[/itex]
Calculating the Jacobian of this gives us [itex]2+4\,{\frac {{y}^{2}}{{x}^{2}}}[/itex]
This is where I'm stuck as I'm not sure how to get this into a form (in terms of u and v) to put in the integral.
I considered using polar substitution, but that seems to be messier.
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