Question

Find a parametric representation for the surface.

The part of the sphere\({x^2} + {y^2} + {z^2} = 4\)that lies above the cone\(z = \sqrt {{x^2} + {y^2}} \)

Short answer

The parametric representation of the surface is

\((r\cos \theta ,r\sin \theta ,\sqrt {4 - {r^2}} )\) where \(\theta \in \left( {0,2\pi } \right)\;\& r \in \left( {\sqrt 2 ,2} \right)\)

Step-by-step solution

Parametric equation and simplification

The parametric representation of the points, inside the cone is

\((r\cos \theta ,r\sin \theta ,z)\) where \(0 < r < z\;\& \;\theta \in (0,2\pi )\)

Draw the graph

The equation of the sphere can be re-written as \(z = \sqrt {4 - ({x^2} + {y^2})} \).

Substitute \(x = r\cos \theta \;\& \;y = r\sin \theta \), to get

\(\begin{aligned}z & = \sqrt {4 - ({r^2}{{\sin }^2}\theta + {r^2}{{\cos }^2}\theta )} \\ & = \sqrt {4 - {r^2}} \end{aligned}\)

Therefore, the points inside the cone and part of the sphere are given by

\((r\cos \theta ,r\sin \theta ,\sqrt {4 - {r^2}} )\) where \(\theta \in \left( {0,2\pi } \right)\;\& r \in \left( {\sqrt 2 ,2} \right)\)

The range of r

To find the range of r, we equate the z coordinates,

\(\begin{aligned}\sqrt {{x^2} + {y^2}} & = \sqrt {4 - ({x^2} + {y^2})} \\{x^2} + {y^2} & = 4 - ({x^2} + {y^2})\\{r^2} &= 4 - {r^2}\end{aligned}\)

\(\begin{align} & 2{{r}^{2}}=4 \\ & {{r}^{2}}=2 \\ & r=\sqrt{2} \end{align}\)