Hi Sally,
Using a mask would do what you want.
Select all the black lines and group them (Ctrl+G).
Then duplicate the white object Ctrl+C (Copy) and then paste Ctrl+F (Paste in place), make sure the the duplicate is at the front.
Then selecting the black lines you can see and the white object/shape that you have brought to the front (Holding shift), press Ctrl+7 (Mask) and you will get the result you want.
Let us know how you get on?
Edit: Quick video to help: