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Homework Statement
Find the volume of the region R between the surfaces [itex]z = 4x^2 + 2y^2 \space and \space z = 3 + x^2 - y^2[/itex]
Homework Equations
The Attempt at a Solution
Okay so I think I have an idea about how to do this one. First I check when the two surfaces intersect, that is when [itex]4x^2 + 2y^2 = 3 + x^2 - y^2[/itex]. So the two surfaces meet when [itex]x^2 + y^2 = 1[/itex] which is a circle of radius 1 with center (0,0).
So either x or y run from -1 to 1, it doesn't really matter which one I pick here I think, so I'll hold y fixed : [itex]-1 ≤ y ≤ 1[/itex]
Now solving [itex]x^2 + y^2 = 1[/itex] for x yields [itex]x = ± \sqrt{1-y^2}[/itex] as my upper and lower limits for x.
Now all that's left to figure out is... what is f? I believe intuitively that f will be the bigger surface minus the smaller surface. So subtracting my two surfaces I get : [itex]3 + x^2 - y^2 - 4x^2 - 2y^2 = 3(1 - x^2 -y^2)[/itex]
So that my iterated integral becomes :
[itex]\int_{-1}^{1} \int_{- \sqrt{1-y^2}}^{\sqrt{1-y^2}} 3(1 - x^2 -y^2) \space dxdy[/itex]
Evaluating that shouldn't be a problem, I'm just hoping I did everything properly while setting it up. If anyone could clarify for me it would be great :)!