Volume of Spherical Cap Section

So,

In calculating the volume of a partially-filled horizontal cylindrical tank which has ends similar to spherical caps, how do you calculate the volume of a partially-filled spherical cap (from the side of a sphere?)??

The cylinder I have solved, and the volume of a full spherical cap has been easy to find but the partial volume of the cap ends is tough.

This has been so hard for me to find information on, so I hope someone here has an idea!

The dimensions of this particular problem are: Cylinder Length = 8.16m, Diameter = 3.7m, Cap "height" (although horizontal) = 0.53m. I hope this is enough detail...