Let the curve C be the boundary of the region bounded by $y=2$ and $x^2+y^2=9$.
a) How would C look like?
(I think it would be circle segment?)
b) Which of the following characteristics does C have?
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smooth
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piece wise smooth
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Jordan curve
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closed
(I think it is closed and smooth and also a Jordan curve?) But how can I prove this mathematically?
c) I need to find a parameterization of the curve, and here I do not know how to start, do I parameterize the line and the circle arc separately?
the intersection points would be $P(\sqrt{5} | 2), Q(-\sqrt{5} | 2)$
Best Answer
For the parametrization, if it is the upper sliver, you can solve the quadratic equation for $y$:
$y=\pm\sqrt{9-x^2}$ and since we’re working in $y>0$, simply $y=\sqrt{9-x^2}.$
Then letting $x=t$ we have the parametrization $(t,\sqrt{9-t^2})$ for $-\sqrt{5}\leq t\leq\sqrt{5}$.
Your line is $y=2$ thus the parametrization $(t,2)$ for $-\sqrt{5}\leq t\leq\sqrt{5}$.
These are easy ways to think of parameterizations at first, you can work your way to different methods such as polar coordinates.