the two consecutive positive integers can be written as x and x+1, where x is a positive integer

the sum of their squares is 181

x^2 + (x+1)^2 = 181

by solving we find

x = 9

click here to see the step by step solution of the equation

9 + 1 = 10

indeed 9^2 + 10^2 = 81 + 100 = 181

the two consecutive positive integers are 9 and 10.