Question about the proper interpratation of integral notation The book I am using defines the integral $I_N = \int_{-N}^{N}f(x)dx$ and then wants to show that if f(x) is a moderately decreasing function then the sequence of these integrals is a Cauchy sequence.
It then states that given M > N, $\quad$ $|{I_M - I_N}| \leq |\int_{N \leq |x| \leq M} f(x)dx|$
I am just a little unsure of how I should read this integral notation involving the absolute value of x being between two integers.
Thinking about it on a 2-d plane it would seem to suggest that $$\int_{N \leq |x| \leq M} f(x)dx = \int_{N}^{M}f(x)dx + \int_{-M}^{-N}f(x)dx$$
is this correct?
 A: You are entirely correct.
In the context of the Riemann integral on $\mathbb{R}$, the notation
$$\int_{N \le |x| \le M} f(x)\,\mathrm{d}x $$
can be a little disconcerting—such integrals are typically defined over intervals in terms of Riemann sums.  In this context, you are correct to presume that if $x$ is a real variable, then
$$ \int_{N \le |x| \le M} f(x)\,\mathrm{d}x
:= \int_{-M}^{-N} f(x)\,\mathrm{d}x + \int_{N}^{M} f(x)\,\mathrm{d}x. $$
More generally, however, we might be better served by considering the integral as being defined over a more arbitrary set, rather than on an interval.  There is far too much theory to get into here, but this is (perhaps) most naturally understood in the context of the Lebesgue integral.  This notion also appears (as you have suggested) in the context of multivariable integration.  For example,
$$ \iint_{R} f(x,y)\,\mathrm{d}A $$
is common notation for a integral (Riemann or otherwise) over a region $R$ in two-dimensional Euclidean space.  In either case, the notation can be interpreted as
$$ \int_{N \le |x| \le M} f(x)\,\mathrm{d}x
:= \int_{R} f(x)\,\mathrm{d}x, $$
where $R$ is the set
$$ R = \{x : N \le |x| \le M\}. $$
