Integral of Dirac Delta Function If the Dirac Delta function has an infinitely tall "spike" at $x=0$, then how is it that its integral over the entire real number line is one, instead of infinity?

To talk about the Delta function rigorously, you need to introduce the idea of distributions. In this case you never look at $\delta(x)$ outside an integral, and you define it to have the property
$$\int\delta(x-x_0)f(x)dx=f(x_0),$$
which then implies it having unit integral if we take $f(x)=1$.
In fact, if we actually define $\delta(x)$ to be $0$ everywhere and $\infty$ at $0$, then its Lebesgue integral should be zero.
We can however look at $\delta(x)$ as the limit of a series of Gaussians:
$$\delta_a(x)=\frac{1}{a\sqrt{\pi}}e^{-x^2/a^2},$$
as on the Wikipedia page. Each Gaussian $\delta_a$ has unit integral, and we can show that as $a\rightarrow 0$, we start to arrive at the properties of $\delta(x)$. You can also see in the animation that it approaches the shape of being zero everywhere, but highly peaked at the origin.
