Your post is a bit muddled but all you need to do is work out

which you already had correct and it does give the answer you quoted.

(-1)^6 = 1 and -2 x -1 =2

so you have 1/6+2= 2 1/6

By the way you normally sub in the upper limit first so in this case the area is negative (below the x axis) but the answer you should give is the absolute value as given in the book 2 1/6.