No good. You started out with a rectangular region. Get rid of the white space below the blue region and above the x-axis.

Work with the lines

y = 1

y = 1 - x ==> x = 1 - y

y = x - 1 ==> x = y + 1

\(\displaystyle \int_{0}^{1}\int_{1-y}^{y+1} c\;dx\;dy\;=\;1\)

Or, to save on postage, you can just look at the geometry. Area = ½*Base*Height = ½*2*1 = 1

Now what?