You need to listen to your own title!!

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