I don't believe you need to use a double integral, just partition the sphere on the interval 0 to the h(depth) of the tank, and to find the volume you calculate the integral of the area on the interval 0 to h I believe, can someone confirm this as I am not 100%

Possible consider

I based this off if the tank was standing on the x-y plane, with y having relation to h and your partitions are being taken from 0 to some point h on the h-axis so to speak which can be related to some depth h, which the change in thickness of your partitions will be Also note to determine your radius it needs to be in terms of h, instead of the usually given function that is in terms of x which remember I stated h has relation to y-axis, since your radius is a, I assume your integral would look like this