UPDATE: I believe the original authors made a mistake. Please see below for more detail.
I came across the following approximation of the integral:
$\int_{a}^{a+b} f(x)dx \simeq f(a)b$.
by assuming $f(x)$ is constant on the interval [a, a+b]. Furthermore, $a,b \in \mathbb{R}$ and $b$ is very small.
My first question is whether the following approximation is equally correct:
$\int_{a}^{a+b} f(x)dx \simeq f(a+b)b$
If not, then can you please explain why the first approximation holds?
My second and main question, however, concerns the approximation of the double integral below.
Suppose now we have a multivariate density function $g(x,y)$ sucht that $f(x) \equiv \int_yg(x,y)dy$.
The variable $x$ now has range $[a, a +b_y]$. Here $b_y$ is increasing in $y$ and $y \in [0,\overline{y}]$. The author assumes $g(x,y)$ is constant in $x$ on the interval $[a, a+b_y]$ and goes on to use the following approximation:
$\int_y \int_{a}^{a+b_y} g(x,y)dxdy \simeq f(a)E[b_y]$
where $E[\cdot]$ is the expectations operator.
Can someone please explain how we arrive at this approximation?
UPDATE: I believe the authors meant to write $E[b_y | X=a]$, but instead write $E[b_y]$. Am I correct? Here is my attempt:
$\int_y \int_{a}^{a+b_y} g(x,y)dxdy \simeq \int_y g(a,y)b_ydy = \int_y \frac{g(a,y)}{f(a)}f(a)b_y dy = f(a) \int_y \frac{g(a,y)}{f(a)}b_y dy = f(a) E[b_y | X=a] $
Is this derivation correct?