Well (1) is obviously 1 as the measure of the set where $\lfloor x \rfloor = 1, ~x \in [1,2]$ is $1$ and $1^x = 1~ \forall x$

The only other value $f(x)$ has on this interval is $f(2)=4$, but the interval $[2,2]$ has measure 0 so adds nothing to the average.

(2) $\bar{a} = \lim \limits_{N\to \infty} \sum \limits_{n=1}^N \dfrac{1}{n} =

\lim \limits_{N\to \infty} \dfrac 1 n H(n) \leq \lim \limits_{N\to \infty} \dfrac{\ln(N)+1}{N} = 0$

To be honest, I find the result in (2) troubling. The infinite sum of positive values produces an average of 0. That doesn't make sense to me, but such is the nature of infinity.