Putnam 1948 B5

The pairs $(a,b)$ such that $|a+bt+ t^2 |\leq 1$ for $0\leq t \leq 1$ fill a certain region in the plane. What is the area of this region?

Dataset text AI-MO AoPS. Quoted with source.

Solutions

No solution is on olymmath yet. Official and human solutions come first when they arrive. AI solutions come after them, with a label.