Is there a way to calculate integrals directly, rather than through an approximation like sums in Riemann integral? Is there a way to calculate integrals directly, rather than through an approximation like sums in Riemann integral?
By "directly" I mean to produce the integral as it is ("analytically") w/o resorting to a method that approximates it through simpler shapes. So e.g. perhaps such method would somehow infer, what an area, volume etc. must be, given what size the boundary has?
I thought that shouldn't Fourier analysis lend itself to the discovery of such method, but maybe not?
 A: The OP asked for a way to interpret integration analytically but without approximating with simpler shapes.
Unfortunately, approximating the integral with Riemann (or Lebesgue) sum is the analytic definition of integral.
The OP also asked whether an integral can be calculated from the length of the curve of the boundary.
The length of the boundary of a curve generally does not depend on its area. Also, the length of the graphic of a function is expressed as an integral $\int_a^b\sqrt{1+f(x)^2}dx$. So, to find the length of a curve you have to find another integral, which is often more complicated.
That said, I assume the OP meant interpreting integration algebraically, rather than analytically, because in that case the question makes sense.
If we interpret integration as an algebraic operation, inverse to formal differentiation, we do not use the analytic, Riemann sum definition of an integral, we just apply formal rules of integration.
For instance, in formal power series there is no sense to talk about the area under the curve.
In that case, integration can be represented as multiplying the power series with this infinite matrix:
$$D^{-1}=\left(
\begin{array}{cccccc}
 0 & 0 & 0 & 0 & 0 & . \\
 1 & 0 & 0 & 0 & 0 & . \\
 0 & \frac{1}{2} & 0 & 0 & 0 & . \\
 0 & 0 & \frac{1}{3} & 0 & 0 & . \\
 0 & 0 & 0 & \frac{1}{4} & 0 & . \\
 . & . & . & . & . & . \\
\end{array}
\right)$$
