The standard choice would be the usual representation in spherical coordinates. Let
[tex]x=4\cos\theta\sin\varphi[/tex]
[tex]y=4\sin\theta\sin\varphi[/tex]
[tex]z=4\cos\varphi[/tex]
We get the part of the sphere between the planes [tex]|z|=2[/tex] with [tex]0\le\theta\le2\pi[/tex] and [tex]\dfrac\pi3\le\varphi\le\dfrac{2\pi}3[/tex].
To Ā answer this question, we should make use of spherical coordinates.
Solution is:
S = ( Ā 4ĆcosĪø ĆsinĪ¦ , 4 ĆsinĪøĆ sinĪ¦, 4 Ć cosĪ¦)
0 ā¤ Īø ā¤ 2ĆĻ Ā Ā ; Ā Ļ/2 Ā ā¤ Š¤ ā¤ (3/2)ĆĻ
In Analitic Geometry we have different way of determine, and identify Ā the position of objects, we have rectangular coordinates, cylindrical coordinates and spherical coordinates. The use of each of these system depends on de geometry of the problem.
In this particular case and according to the problem statement we should use spherical coordinates
x = ĻĆcosĪø ĆsinĪ¦ Ā Ā Ā Ā Ā y = Ļ ĆsinĪøĆ sinĪ¦ Ā Ā Ā Ā z = ĻĆ cosĪ¦
In our particular case
Ļ = 4 Ā then Ā x = 4ĆcosĪø ĆsinĪ¦ Ā Ā y = 4 ĆsinĪøĆ sinĪ¦ Ā z = 4 Ć cosĪ¦
0 ā¤ Īø ā¤ 2ĆĻ Ā Ā ; Ā Ļ/2 Ā ā¤ Š¤ ā¤ ( 3/2)ĆĻ
So the solution in terms of Īø and/or Ā Ī¦
S = ( Ā 4ĆcosĪø ĆsinĪ¦ , 4 ĆsinĪøĆ sinĪ¦, 4 Ć cosĪ¦)
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