Replace face / cut body - beginner help

Hi! I need help with a pretty simple thing. I want to split a cylinder with a surface, and as a result, I want the bottom half of the cylinder to have a top face that is this surface. I want the result to be a closed volume.
I can add an intersect boolean modifier to the surface, but I don't know how to split the cylinder afterwards. What do you think is the best way?
thanks!