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Volume using spherical coordinates!

Hi tsr,

Let V be the region in R3 defined by the inequalities z^2≥x^2+y^2, x^2+y^2+z^2≤1, and z≥0. Sketch the region V and find its volume?

I have sketched V but to find the volume I have written the region in spherical coordinates but I don't understand how to determine the value of theta and phi? I know what theta and phi are for a point in space but not how to find them for a region in space.

Also how do you know when to use spherical coordinates and when to use cylindrical ones?

Any help would be much appreciated.
Reply 1
Avatar for BlueSam3
BlueSam3
17
Integrate over the region using the volume element from your coordinate system, using whatever limits are easiest.

Use whichever seems easiest, or if they're both awkward, make up your own coordinate system that makes it easier, having computed the area element for that system.
Reply 2
Avatar for gn17
gn17
OP
1
Original post by BlueSam3
Integrate over the region using the volume element from your coordinate system, using whatever limits are easiest.

Use whichever seems easiest, or if they're both awkward, make up your own coordinate system that makes it easier, having computed the area element for that system.


I have worked out the limits of phi but don't understand why the limits of theta are 0 to pi/4 (in the m.s) as i get them to be 0 to pi/2.
Original post by gn17
I have worked out the limits of phi but don't understand why the limits of theta are 0 to pi/4 (in the m.s) as i get them to be 0 to pi/2.


I presume theta is the angle with the positive z axis?

Try drawing a cross section through the plane y=0, for example, and all should become clear.
(edited 10 years ago)

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