I will give this a go using a triple integral. I will give my 2 cents. If I have an epiphany, I will certainly be back.
What we have is half a cylinder which creates the semi circle.
Keep the tank horizontal and look at the water surface as a tilted plane. Take
the tank as the cylinder , which goes
along the z-axis. I suppose it doesn't matter which axis you use though.
Then, calculate the volume by integrating with respect to z, with the sections
perpendicular to the z-axis. The sections are segments of a circle.
I would think it would depend on two different scenarios. One with the top
end(the end raised in the air) being partially covered and then the case when
the water is down enough that the top end is dry.
Here is an integral I came up with. Admittedly, I am not 100% comfortable. See what you think.
I will look again later. I have a feeling we may need another integral for the other case. If you find anything, let me know. Cool problem.