volume using triple integrals in spherical cordinates

[aron101782]

here is a wedge cut out of a unit sphere
[MATH]\int_{0}^{\frac{\pi}{6}}\int_{0}^{\pi}\int_{0}^{1} \kappa^2\sin\theta d\kappa d\phi d\theta=\frac{\pi}{9}[/MATH]I would like to know how to get this same wedge only rotated 90 degrees such that theta is along the axis and phi is along the positive y axis. I have tried the following:
[MATH]\int_{0}^{\pi}\int_{0}^{\frac{\pi}{6}}\int_{0}^{1} \kappa^2\sin\theta d\kappa d\phi d\theta=\frac{\pi}{3}(1-\frac{\sqrt{3}}{2})[/MATH]which does not cut the same shape. My question is how get that same cut from the top of the sphere preferably in spherical coordinates.
im not sure why the latex is messed up /kappa

 

[LCKurtz]

here is a wedge cut out of a unit sphere
[MATH]\int_{0}^{\frac{\pi}{6}}\int_{0}^{\pi}\int_{0}^{1} \kappa^2\sin\theta d\kappa d\phi d\theta=\frac{\pi}{9}[/MATH]

Including a picture and defining your variables would help. Is [MATH]\kappa[/MATH] the radius usually called [MATH]\rho[/MATH]? And is [MATH]\theta[/MATH] the usual polar coordinate angle in the xy plane? And [MATH]\phi[/MATH] the usual spherical coordinate azimuth angle?

I would like to know how to get this same wedge only rotated 90 degrees such that theta is along the axis and phi is along the positive y axis.

I can't make sense of that. A sketch would be helpful. I don't know what [MATH]\theta[/MATH] "along the axis" means. Ditto [MATH]\phi[/MATH] "along the y axis".