I have an assignment problem which asks me to decide if the regions provided are horizontally or vertically simple. I am stuck on one of these which is the following:
$$ A = \{(x,y)\in \Bbb R^2 | 2 \le x^2 + 2y^2 \le 4\}$$
I have the definition and sketches from lectures of what it means for a region to be horizontally or vertically simple however I do not understand how to apply those to the given set.
In class we did double integrals of vertically and horizontally simple regions and changed the limits on the integrals such that one integral's limits were a function of x or y. In this case I am not really sure what to do. I have tried a few things but am completely lost.
Thank you for any suggestions.
$\endgroup$1 Answer
$\begingroup$Here is our area.
$x^2+2y^2\leq4$ corresponds to the area inside of an ellipse (outer ellipse in the picture). $2\leq x^2+2y^2$ carves out another one, smaller ellipse, from the first one.
So we see, that $A$ is neither vertically simple, nor horizontally simple.
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