Question

how to approximate the square root to the nearest integer

Answer 1

Take the nearest two perfect square root and decide from there. For example.
sqrt 85 is between sqrt 81 and sqrt 100.
sqrt 81=9
sqrt 100=10
The integer of sqrt 85 must be between 9 and 10; since 85 is closer to 81 than it is to 100, you would say sqrt 85 is approximately 9.