Explain your thinking process to find the sum of all integers between 12 and 40

You can treat it as an arithmetic sequence, so multiply the average integer value by the number of integers:
  (40+12)/2 × (40 -12 +1) = 26×29 = 754

You can subtract the sum of integers 1–11 (66) from the sum of integers 1–40 (820) to get the sum of integers from 12–40: 820 - 66 = 754.

You can let a graphing calculator do it. (See attached.) This method takes perhaps the least amount of thought. It, too, gets a sum of 754.