An ellipse
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ellipse
If it is symmetrical, it could be an ellipse, but there are very many possible answers.
No, it would not.
Your question seems very confused. The normal convention of the Cartesian coordinate system would place negative numbers below the x axis, so that any curve approaching negative infinity would curve downward, not upward.
Take the definite integral (and your bounds should be the two places where the curve crosses the x-axis).