Spherical, Cyndrical or Polar Coordinates

[Northbysouth]

Spherical, Cylindrical or Polar Coordinates

Homework Statement

I have attached an image of the problem.

I know that the solution is number 1 but I'm having some difficulty understanding why. In solution one is it using cylindrical coordinates> My first response to this question had been to use spherical coordinates. Would it have been correct to use spherical coordinates if the \sqrt{}2-x^2-y^2 had instead been \sqrt{}1-x^2-y^2

Any input would be greatly appreciated.

Homework Equations

The Attempt at a Solution

I know that the solution is number 1 but I'm having some difficulty understanding why. In solution one is it using cylindrical coordinates? Because my first response to this question had been to use spherical coordinates. Would it have been correct to use spherical coordinates if the \sqrt{}2-x^2-y^2 had instead been \sqrt{}1-x^2-y^2

 

[haruspex]

Two forms are offered using spherical coordinates, two using cylindrical. In principle, any or none of them could have been correct reformulations of the original integral, but they have been constructed so that only one is correct. You just have to play around converting the original various ways until you can decide which one.
The nature of the z bound in the original integral does suggest spherical as the most natural, but that's not what the question is about.

 

[HallsofIvy]

Homework Statement

I have attached an image of the problem.

I know that the solution is number 1 but I'm having some difficulty understanding why. In solution one is it using cylindrical coordinates> My first response to this question had been to use spherical coordinates. Would it have been correct to use spherical coordinates if the \sqrt{2-x^2-y^2} had instead been \sqrt{}1-x^2-y^2

It should be easy to see that z= \sqrt{1- x^2+ y^2} is the upper half of a sphere with center at (0, 0, 0) and radius 1 while z= \sqrt{2- x^2+ y^2} is the upper half of a sphere with center at (0, 0, 0) and radius \sqrt{2}.

Because they are both parts of spheres, yes, spherical coordinates would be appropriate.

Any input would be greatly appreciated.

Homework Equations

The Attempt at a Solution

I know that the solution is number 1 but I'm having some difficulty understanding why. In solution one is it using cylindrical coordinates? Because my first response to this question had been to use spherical coordinates. Would it have been correct to use spherical coordinates if the \sqrt{}2-x^2-y^2 had instead been \sqrt{}1-x^2-y^2

Homework Statement

Homework Equations

The Attempt at a Solution