TITLE: The negative integral meaning
QUESTION [13 upvotes]: Whenever I take a definite integral in aim to calculate the area bound between two functions, what is the meaning of a negative result? Does it simly mean that the said area is under the the x - axis, in the negative domain of the axis?

REPLY [2 votes]: This area thing is for kids. What you have to grasp is the general idea of an integral.
We are given a solid $B\subset{\mathbb R}^n$, $n\geq1$,  and a function $f:\>B\to{\mathbb R}$ modeling some variable (geometrical, physical, etc.) intensity on $B$. The integral $\int_B f(x)\>{\rm d}(x)$ then encaptures the total impact caused by $f$ on this $B$ in terms of  a single number. Thinking about this idea, and the properties this total impact should naturally have, brings us to the following setup: Partition $B$ into a large number  of tiny subdomains $B_k$ $(1\leq k\leq N)$ and choose a sample point $x_k$ in each $B_k$. Then the quantity we have in mind is approximately given by
$$\sum_{k=1}^N f(x_k)\>{\rm vol}(B_k)\ ,$$
so that we should define
$$\int_B f(x)\>{\rm d}(x):=\lim_{\ldots}\sum_{k=1}^N f(x_k)\>{\rm vol}(B_k)\ ,$$
whereby the $\ldots$ under the $\lim$ will have to be filled with lots of legalese.
In the simple case of an $f:\>[a,b]\to{\mathbb R}_{>0}$ the function $f$ models an "area intensity" over the interval $[a,b]$, and the integral then is the "area under the curve".