Calculate Asymptotics of Integral?

Solution 1:

Let's view this from the perspective of probability theory (which your profile indicates you are studying).

Hints:

1. $[0,1]$ is compact and $f$ is continuous. What does that tell you about boundedness of $f$?
2. Note that if $U_1, U_2,\ldots,U_n$ are iid uniform random variables on $[0,1]$, then $$ \mathbb E f(\overline U_n) = \int_{[0,1]^n}f\left(\frac{u_1+\cdots+u_n}{n}\right)\mathrm d u_1 \cdots \mathrm d u_n$$ where $\overline U_n = \frac{1}{n} \sum_{i=1}^n U_i$. So, we have a probabilistic interpretation of the integral in question.
3. Argue that $\overline U_n \to c$ (almost surely) for some $c \in \mathbb R$. Identify this $c$.
4. Use continuity of $f$ to conclude that $f(\overline U_n) \to f(c)$.
5. To conclude that $\mathbb E f(\overline U_n) \to a$ for some $a \in \mathbb R$ that you should identify, use hint 1 and any one of a couple different standard results that you should know about convergence of integrals (expected values!) given convergence of a sequence of functions (with appropriate properties).

Finally note that we've made no additional assumptions about smoothness of $f$ while carrying out this program.