Integration with infinite/finite limits as a form of summation in DSP notation for discrete signals While reading some DSP literature, I've noticed integrals of the following form.  An example integral (Equation 7.17) is given on pg. 124 of a book that I've found very challenging to read (Seismic Inverse Q filtering): 

$u(t) = h(t)\int\limits_{ - \infty }^\infty  {u(\tau ,t)d} \tau$

This is exactly how the author has written the equation.  Gauging the author's intent, I think that $u(\tau ,t)$ is really another function, so in the above, let's re-write the equation as the following:

$u(t) = h(t)\int\limits_{ - \infty }^\infty  {p(\tau ,t)d} \tau$

In the above, $u(t) \neq p(\tau ,t)$, so I think that this clears up some of the notation.

This appears to be notation associated with a continuous signal $p(\tau, t)$ that is of infinite length.  For a discrete signal $p[\tau, t]$ with a finite length $N$, how do I evaluate the integral?

Case #1

Is the integration treated as a summation?

$u[t] = h[t]\sum\limits_{i = 1}^N {p[i,t]}$

Case #2

Is the integration treated as a "classical" integration using the trapezoidal rule?

How do I know which of these cases apply for the discrete signal case?  Is this really an issue of notation?

Another Example

Another example can be found on pg. 128 of the same book (Equation 7.21):

$\tilde U(\tau ,\omega ) = U(\tau ,\omega )\frac{1}{{\Lambda (\tau ,\omega )}}\exp \left[ {i\int\limits_0^\tau  {\left( {{{\left( {\frac{\omega }{{{\omega _h}}}} \right)}^{\gamma (\tau ')}} - 1} \right)} \omega d\tau '} \right]$

When the integration is being done over $\tau'$, is this integration a summation or a "classical" integration?

Yet another example

On pg. 9 of  "R. Wang, Introduction to Orthogonal Transforms: With Applications in Data Processing and Analysis. Cambridge University Press, 2012", the author states that the energy in a continuous signal $x(t)$ is given by:

$E = \int\limits_{ - \infty }^\infty  {{{\left| {x(t)} \right|}^2}} dt$

The energy in a discrete signal $x[n]$ is:

$E = \sum\limits_{n =  - \infty }^\infty  {{{\left| {x[n]} \right|}^2}}$

But shouldn't a discrete signal have finite lower and upper bounds of summation?  What is really meant here using infinite lower and upper bounds?  Shouldn't the trapezoidal rule be used here?

So here is the gist of my question: Using this notation, is integration done using the trapezoidal rule for discrete signals, or is it simply a form of summation?  What is the general use of this type of notation in DSP?

Given an equation for a continuous function integration, how does this relate to integration in the context of a discrete function?
In the absence of any further information, we must assume that tradition plays a role in the numerical implementation.  The language of mathematics implies that an integration is an integration, so we can assume that the trapezoidal rule holds:

$\int_a^b f (x){\mkern 1mu} dx \approx \frac{1}{2}\sum\limits_{k = 1}^{N} {\left( {{x_{k + 1}} - {x_k}} \right)} \left( {f({x_{k + 1}}) + f({x_k})} \right)$

However, the intent of the author is another mitigating factor, so careful reading is required.  There are often some pathological issues of notation, but these are strange.  

After some numerical experimentation, I've found that in general, an integration is an integration, no matter which field of inquiry.  The language of mathematics is universal.