Suppose I have an integrable function $f(x)$, which I haven estimated from data. For every $x$ in $\mbox{domain}(f)$ I have an estimate of the variance $\mbox{Var}(f(x)) = \sigma^2_x$.
Suppose furhter I wanted to integrate $f$ over a subset of its domain. Could I obtain an estimate of the variance of the area under $f$?
If every element in $\mbox{domain}(f)$ is independent, I suppose the integral would look like
$$ I(f) = \int_a^b f(x) \, dx \approx \dfrac{b-a}{n} \sum_n f(x_n)$$
So then
$$ \mathbb{E}(I(f)) = \dfrac{b-a}{n} \sum \mathbb{E}(f(x_n))$$
and
$$ \mbox{Var}(I(f)) = \left( \dfrac{b-a}{n} \right)^2 \sum_n \sigma^2_{x_n} \>.$$
Is that correct? If so, in the limit as $n$ grows arbitrarily large, what does the variance look like?