If it is symmetrical, it could be an ellipse, but there are very many possible answers.
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An ellipse
ellipse
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).