i suspect you are not telling us the whole problem, just a part of it. your answer for the definite integral is correct, and matches what you were given:
2 - 8√2/3 = 6/3 - 8√2/3 = (6 - 8√2)/3
it looks if you're using an integral version of the mean-value theorem, that is, c is a value in (0,2) where the definite integral of f from 0 to 2 matches a rectangle of height f(c) and width 2 (the width of the interval [0,2]).
if we define , for x in [a,b] then by the mean-value theorem, there is a c in (a,b) with: