In this case can I break the mod value and write like this
Indeed...using just "plain geometry" (ok, a lame pun), it is apparent that the integral:
, while:
.
The reason being, the triangle with base extending from (-2,0) to (-1,0) in the former integral has a "peak" at (-2,-1) and lies BELOW the x-axis (so has "negative area", that is the integral is negative in this region), while in the latter integral lies above the x-axis (with "peak" at (-2,1)). This triangular area has a base of 1, and a height of 1, and so has area 1/2, making a "swing" of a difference of 1 between the two integrals.
And yes, Halls, that WAS a typo. :P