Boundary
Zone
What shows the district and region and where they exist? That would be the boundary. It is what surrounds every district and region, showing where it exists.
くいき
This is a jukugo word, which usually means on'yomi readings from the kanji. If you know the readings of your kanji you'll know how to read this as well.
日本政府は、その区域の正確な大気中の放射能濃度を公表しているのですか？
Will the Japanese government announce the exact zone in the atmosphere where the radioactivity is concentrated?