We find the sum of all positive even integers less than 233 and then subtract all the positive integers less than 233 that are divisible by 10. $2 + 4 + \ldots + 232 = 2(1 + 2 + \ldots + 116) = 116 \cdot 117 = 13572$. The sum of all positive integers less than 233 that are divisible by 10 is $10 + 20 + \ldots + 230 = 10(1 + 2 + \ldots + 23) = 2760$. Then our answer is $13572 - 2760 = 10812$.