Intuitive meaning of more than one Integral If the integral of a function represents the area beneath the curve represented by that function, what do double or triple integrals represent?
For example, what do the following integrals represent intuitively: 
(1)    $\int_a^b \int_c^d f(x, y) dx dy$
(2)    $\int_a^b \int_c^d\int_e^f f(x,y,z) dx dy dz$
 A: If $B\subset {\mathbb R}^n$ is a reasonable $n$-dimensional domain, and $f:\>B\to{\mathbb R}$ is some variable "intensity" defined on $B$ then the integral $$\int_B f(x)\>{\rm d}x$$captures the total impact these givens create. The idea is that we should have
$$\int_B\bigl(f(x)+g(x)\bigr)\>{\rm d}x=\int_B f(x)\>{\rm d}x+\int_B g(x)\>{\rm d}x\ ,\quad \int_B \lambda f(x)\>{\rm d}x=\lambda \int_B f(x)\>{\rm d}x\ ,$$
and
$$\int_{B_1\cup B_2} f(x)\>{\rm d}x=\int_{B_1} f(x)\>{\rm d}x+\int_{B_2} f(x)\>{\rm d}x$$
if $B_1$ and $B_2$ are essentially disjoint, and finally that
$$\int_B 1\>{\rm d}x={\rm vol}(B)\ .$$
When $f$ is continuous then these requirements lead to the insight that such an "integral" has to be the limit of Riemann sums:
$$\int_B f(x)\>{\rm d}(x)=\lim_{\ldots}\ \sum_{i=1}^N f(\xi_i){\rm vol}(B_i)\ ,$$ where the $B_i$ form a partition of $B$ into small subdomains.
The integrals you have exhibited refer to the special case where $B$ is a rectangle, resp., a rectangular box, and to the way such integrals are actually computed when $f$ is given as an expression in the variables $x_1$, $\ldots$, $x_n$.
A: By Fubini's theorem you can integrate on a first variable, then on a second...
$$\int_a^b \int_c^d f(x, y) dx dy= \int_a^b F(y) dy
$$
is thus the integral over $y$ of the area under the curve $z=f(x,y)$ for fixed $y$.
This is nothing but the volume under the surface $z=f(x,y)$.
A: Double and triple integrals have the same idea but in more dimensions. I suggest you think of it a bit like mappings. 
For double integrals you are mapping a 2D surface (namely the xy plane) by a function to another surface, and the distance between those two surfaces at every point is evaluated giving the volume
Triple integrals are a bit more difficult to understand visually but also share the same concept. You have a function which 'maps' a 3D object to another 3D object and calculates the 'distance' between them.
In fact if you set the function to be equal to 1, you find the area of the region you're double integrating and the volume of the region in triple integration. 
It is the same reason why if you have a rectangle with one of its sides equal to 1, then the area of the rectangle is exactly equal to the remaining side numerically. 
