Originally Posted by

**Deveno** 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 $\displaystyle F(x) = \int_a^x f(t)\ dt$, for x in [a,b] then by the mean-value theorem, there is a c in (a,b) with:

$\displaystyle f(c) = F'(c) = \frac{F(b) - F(a)}{b - a}$.

that is:

$\displaystyle f(c)(b - a) = \int_a^b f(t)\ dt$