I indicated a graph for $\theta, r$ of demarcation line in the above sketch. ![]() You can appreciate the relationship between $r, z $ and $\theta$ using chosen parameter $u$, if you dont want to consider change of independent variable to $u$. " CYL COORDS ON ELLIPTIC PARABOLOIDS SE NOV 2014" Hence, we can use our recent work with parametrically defined surfaces to find the surface area that is generated by a function f f ( x, y) over a given domain. $$4y^2 x^2=4\Rightarrow 4r^2\sin^2\theta r^2\cos^2\theta = 4 \Rightarrow r=\frac,GridLines->Automatic] As we noted earlier, we can take any surface z f ( x, y) and generate a corresponding parameterization for the surface by writing. Question Using the double integral for polar coordinates find the area. Also I would like to know the equation of the curve of intersection between the cylinder and the paraboloid, but don't know how to do it.įor the $r$ bounds I tried considering the equation for the base: Find the surface area of the part of the circular paraboloid zx2 y2 that. However I'm having difficulties in determining the $r$ bounds. So I take the transformation to cylindrical coordinates: Here there's a horrible sketch of the solid: However I have never dealt with a problem in which $r$ bounds depend on $\theta$. I would like to use cylindrical coordinates.
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