Intuition for actions written as integrals over spacetime Right now I'm simply looking for an intuitive explaination of actions that integrate over a 4-volume element, $d^4x$ rather than a parameter say $\lambda$.  More specifically I'm well versed in action principles that say have an action as $\int L d\lambda$. But in trying to understand actions such as the Einstein Hilbert action for example that are an integral over a volume element, in the form $\int(whatever)d^4x$. I'm not exactly seeing from an intuitive stance how these are pieced together, and how/if they relate to the extremization methods I am more familiar with. I do understand the notation of metrics, and GR and all that, that's not the problem. I just don't have any experience working with actions that are integrals over volumes. If anyone can try to point me in the right direction here, or recommend a good text perhaps it would help.
 A: Let's compare classical mechanics and GR to attempt to get at the intuition you're looking for.
Classical mechanics.
Recall that in the classical mechanics of a system of $N$ particles, the configuration of the system at every point in time is represented by a point $x\in\mathbb R^{3N}$.  The configuration manifold $\mathcal Q$, namely the set of possible configurations, is actually usually a submanifold of $\mathbb R^{3N}$ because there are some constraints.  
The objective of classical mechanics is to predict the motion of the system on its configuration manifold for all times $t$ given the configuration and velocity of the system at some initial time $t_0$.  
In other words, the objective is to determine a curve $x:[t_0, \infty)\to \mathcal Q$ that represents the motion of the system given the initial data.  In classical mechanics, this can be accomplished via an action principle.  Namely, there exists a functional $S$ that maps curves $x:[t_a, t_b]\to\mathcal Q$ on configuration space to real numbers such that the motion of the system is governed by equations of motion that result from requiring that motions of the system are stationary points of the action which is typically an integral over a local Lagrangian;
\begin{align}
  S[x] = \int_{t_a}^{t_b}dt\, L(t, x(t), \dot x(t))
\end{align}
Another way of saying this is that if $\mathcal C$ denotes the set of all admissible curves on $\mathcal Q$, then the action is a functional $S:\mathcal C\to \mathbb R$ whose stationary points are physical motions of the system.
General relativity. 
In general relativity, instead of wanting to solve for the motion of a system of particles on a given manifold, one often wants to solve for the metric of spacetime given some other information.  For example, perhaps you know you're looking for a static, spherically symmetry spacetime.  One could then consider the set of all admissible metrics, call it $\mathcal G$, and one could ask

Which of the metrics in $\mathcal G$ are physical?

Here $\mathcal G$ is analogous to the set of all curves $\mathcal C$ in classical mechanics.  It turns out that the answer to this question, at least in the context of Einstein gravity without matter and ignoring some subtleties, is that the metric must be a stationary point of the Einstein-Hilbert action;
\begin{align}
  S_\mathrm{EH}[g] = \int_M d^4x\sqrt{|\det(g_{\mu\nu})|} R_g
\end{align}
where $R_g$ is the Ricci scalar for the metric $g$, and $M$ is the spacetime manifold.
The main inuition.
To get intuition for this, recall that in the case of classical mechanics, we were looking for a curve that is a stationary point of $S:\mathcal C\to \mathbb R$, and curves are functions of time, so the action is an integral over time.  In the case of $GR$, we area looking for a metric that is a stationary point of $S_{\mathrm{EH}}:\mathcal G \to \mathbb R$, and metrics are functions of spacetime, so the action is an integral over spacetime.
