What is an integral other than "the area under the graph"? What is a possible visual meaning of the integral of a real function $x(t)$
other than "the area under the graph"? 
I'm asking this so that i can avoid thinking about a graph when thinking 
about an integral, and view the integral as a property of a point in 
infinite-dimensional space.
In the discrete case of a real sequence $x(n)$
there is the sum, finite or infinite:
$x(1) + x(2) + \ldots  =  x(1)*(1-0) + x(2)*(2-1) + \ldots$ 
So also, what is the visual meaning of this sum, which, 
when generalized to a continuous variable, we get the integral.
 A: $f(b)=f(a)+\int_{a}^{b}f^{'}(t)dt$ 
In other words "a later value of the function as the sum of all small changes from another earlier value!"
A: Accumulate acceleration to get speed.
Accumulate speed to get distance.
Accumulate width to get area.
Accumulate area to get volume.
A: In intuitive terms, an integral can be understood as the average of a function over an interval, or more generally over a domain, times the extent of that domain.
E.g. the area under a curve is the width of the interval times the average "height" of the function.
The gravity center of a 3D shape is the average position of the points, i.e. the integral of the position vector over the volume.
The displacement of a vehicle over a time interval is the average speed times the delay, i.e. the integral of the speed.
An integral accumulates instantaneous variations to yield a global variation.
A: Whenever $f$ is the density of some substance, the integral of $f$ is the total amount. This interpretation works well for multiple integrals too.
For example, if $f(x)$ is the electric charge density (charge per unit of length) in a rod $a \le x \le b$, then $\int_a^b f(x) \, dx$ is the total amount of electric charge in the rod. 
And if $f(x,y)$ is the electric charge density (charge per unit of area) in a plate of shape $D$ (a domain in $\mathbf{R}^2$), then $\iint_D f(x,y) \, dxdy$ is the total amount of electric charge in the plate.
And if $f(x,y,z)$ is the electric charge density (charge per unit of volume) in a solid of shape $E$ (a domain in $\mathbf{R}^3$), then $\iiint_E f(x,y,z) \, dxdydz$ is the total amount of electric charge in the solid.
Electric charge is nice since it can be both positive and negative. If $f$ is positive, you can (for example) think of mass density instead.
