From The Three Ages of Pi, can you state your own expression for pi based on Archimede's method?

Again from The Three Ages of Pi, can you show that pi is equal to the integral based on Newton's formula:

In the section entitled Surprising new ideas..., Dr. Borwein discusses the significance of the Bailey-Borwein-Plouffe Pi Algorithm as a digit extraction algorithm. Can you describe a digit extraction algorithm that would return the nth digit in the decimal expansion of any given rational number?

In the fourth section, Dr. Borwein discusses recent records and significant sequences. Do you think we finally have an idea of what the question is?