From Wolfram Alpha, we can sketch the curves to find the area of interest:
Note that we need to find the points of intersection: at $x = 0$ the lines $y = x, \;y = \frac x{4}$ intersect. At $x= 1$, the lines $y = x$ and $y = \frac 1x$ intersect. At $x = 2,$ the lines $y = \frac 1x $ and $y = \frac x4$ intersect. You can solve this by integrating between the relevant curves from $x = 0$ to $x = 1$, and likewise integrating between the relevant curves between $x = 1$ and $x = 2$, then summing:
$$\int_0^1 \left(x - \frac x4\right)\,dx \quad + \quad \int_1^2 \left(\frac 1x - \frac x4\right)\,dx $$
It is a reasonable question. Although usually if asked to find the area of a region or regions bounded by two graphs what is meant by "bounded" is that the regions all lie within the interior of some circle.
This is analogous to a bounded set on the number line being contained in some interval $[a,b]$. It is completely circumscribed.
However it is possible for to graphs to enclose a finite, yet unbounded region.
There are many examples, but one is as follows.
Find the area of the region "bounded" by the graphs of $y=0$ and $y=\dfrac{x}{x^4+1}$
Here is the graph of the region.
This region is not bounded in the sense stated above. It cannot be contained in the interior of a circle. Yet it has a finite area.
\begin{equation}
\int_{-\infty}^\infty\dfrac{|x|}{x^4+1}\,dx=\int_{0}^\infty\dfrac{2x}{x^4+1}\,dx\\
\end{equation}
Make the substitution $u=x^2$, $du=2x\,dx$ and this becomes
\begin{eqnarray}
\int_{0}^\infty\dfrac{1}{u^2+1}\,du&=&\frac{1}{2}\arctan(u){\Large\vert}_{0}^\infty\\
&=&\left(\dfrac{\pi}{2}-0\right)\\
&=&\frac{\pi}{2}
\end{eqnarray}
Therefore it is acceptable to say that, in a sense, an unbounded region is "bounded" by two graphs so long as the area enclosed is finite.
Best Answer
You can rewrite this as an improper integrals:
$$\int_{-\infty}^\infty \left({x^2 + 4\over 4} - {x^2\over 4}\right) \,\mathrm dx = \int_{-\infty}^\infty 1\,\mathrm dx$$
It becomes obvious this does not converge thus the area is not finite:
$$\lim_{b\to\infty} x \,\Big|^b_{-b} \rightarrow \infty - (-\infty) = \infty$$
The notion that the area is undefined because the curves do not cross is wrong. Consider the Gaussian Integral which is between the functions $f(x) = e^{-x^2}$ and $g(x) = 0$. They do not cross, yet the integral from negative infinity to positive infinity is finite:
$$\int_{-\infty}^\infty e^{-x^2} \,\mathrm dx = \sqrt{\pi}$$
Note that $g(x) = 0$ is technically a curve. Mathematically speaking, a curve is a generalization of a line.