You can't both ignore the legend and notes and still interpret the map. The map is of viking settlements in Ireland around 850 AD. The area ringed with yellow dots shows the area that was under viking control during that time. It would be completely irrelevant to "the pale" as designated in 1541. Except for the general shape of Ireland and the location of Dublin, Wexford and Waterford.

The professor is indeed pointing at an area that would be part of "the pale" in 1541. Though he chose a very poor map to use to represent it. Here is a much better map.

If you are actually discussing Vikings and Ireland in 850 the term "The Pale" would be relevant, if slightly confusing, to describe areas solidly under viking control. In modern usage "the Pale" can be used to describe any controlled or "civilized" area surrounded by a less civilized or anarchic area which would be "beyond The Pale."

Erm, no. I'm a little concerned that your professor has chosen a map of Viking Age era Ireland for what appears to be a slide concerning Tudor-era Ireland. The Pale was the hinterland of Dublin that was directly controlled by the English monarchy before its systematic reconquest undertaken by the Tudors. The Pale differentiated land where English law and authority were directly recognized from the more or less autonomous Hiberno-Norman lords who mostly acted independently in their own interests and used English and Irish law depending on the circumstances and the indigenous Irish kingdoms who did not recognize the authority of the English monarch or its laws.

The English monarchy only established direct authority over Munster under Elizabeth, after the Hiberno-Norman Lord of Desmond rebelled twice against the establishment of more systematic methods of royal control in his earldom. This wasn't an expansion of the Pale, but part of a wider process of systematic reconquest of Ireland that was started by Henry VIII and continued by Elizabeth.