You are NOT using spherical coordinates!
Waring 1: Depending on which text you are using and your prof. the representation of spherical coordinates will be different here I will use
Where is the plane angle and is the angle measured from the positive z axis.
So in Spherical coordinates the integrand is
why don't we need an absolute value?
Now Sketch the region of integration and you will see the it is the top half of a sphere of radius 1.
Now to trace a sphere is radius 1
Don't forget to change the differential of volume to spherical coordinates as well.
Fixed my typo above Thanks Ackbeet