Can Laplace Transform be understood as "Area under the curve"? Since Laplace Transform is basically a definite integral of multiplication of two functions $f(t)$ and $e^{-st}$. Can we interpret Laplace Transform as the area under the curve $f(t)e^{-st}$ from $-\infty$ to $\infty$ ?
For example, we know Laplace Transform of $f(t)=t$ for $t>0$ is equal to $F(s)=\frac{1}{s^2}$. Can we interpret this graphically?
 A: I think the best way is to think in analogy to Fourier  transforms. In Fourier transforms you think as your time dependent signal as a superposition of simple sinusoidal functions with different frequencies. This works well for periodic signals. The Laplace transformation is used to describe transitions. You turn on something at $t=0$ and wait to equilibrate, at $t=\infty$. You can write your time dependence as a superposition of exponential functions with different decay rates.
A: Laplace Transform is a generalization of Fourier transform in the sense it can handle a much wider applications in Engineering, Pure and Applied Math. The big difference it can handle signals which grows unbounded in time such as step, ramp or quadratic etc. Another feature of Laplace transform is it can readily solve Initial Value Problem (IVP) while yield Fourier transform for steady state solution (SSS) as a special case when s lies on the jω axis.
The class of functions which admits Fourier Transforms are absolutely integrable.
f ε $L_1 or L_2$.
By contrast functions which admits Laplace Transforms are of exponential order.
i.e. there exists an α such that  |f(x)| <= M exp (αx) as |x|--> ∞
α is (aka) abscissa of convergence for Laplace transform.
Since Laplace transform is suitable for time domain analysis, it is a great tool
for step, ramp and Parabola inputs.  In the S domain, since it includes both the LHP and RHP, we can use Bode & Nyquist plot to see if we have undesirable modes(eigenvalues).
Any eigenvalues in the RHP spells potential trouble for the system designer.  Their first order of business is to see if they can relocate them to LHP so that the system become stable.(Pole Placement).
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In what sense does the Laplace Transform give components of a signal in something called the 's-domain'?
In what sense does the Laplace Transform give components of a signal in something called the 's-domain'?
