I have an oval I am trying to slice two sides from. I select the oval, I add boolean, I select the cube, I select difference and it slices the opposite of what I want leaving me 2 skinny strips and NOT the sliced oval. I've used boolean many times with correct results.. except this time!!
First select both of the objects you are going to use to cut, and join them as a single object (CtrlJ:
Then set the boolean to intersect:
To see the results you can move the cutting object to a separate layer or hide it.
Just for fun, here are the other possible booleans with these two objects: