Question

Is it possible for an unbounded region to have a finite area? Can you give me examples.

Answer 1

Of course just think about the graph of the function y=1/x^2 from x=1 to infinity. The area below the graph is \[\int\limits_1^\infty \frac{1}{x^2}\textrm{d}x=[-1/x]_1^\infty=1.\] So it has a finite area but it' not bounded. See the attached picture: