Gardening and Horticulture - Removing tree roots from lawn

I found out what has been slowly killing a section of my lawn. There are several large tree roots (3-4" diameter) that have spread into this section of lawn. I chopped out about 20' of the roots extending from the tree.

Will the roots that I disconnected from the tree die off on their own? Do I have to trace them down and remove the entire root?

Not worried about losing the tree, I have planted a japanese maple next to it to eventually replace it.

Thanks.

Shallow rooted trees compete with grass for moisture and nutrients. The grass usually loses the contest. If wanting to remove roots from lawn, this will require digging and chopping to eliminate them. If possible, go ahead and remove the tree and provide protection for the Japanese Maple. The tree will compete with the newly planted Japanese Maple for moisture and nutrients.