This is the first part of a long exercise:
We consider heat conduction in a rod of length $l$. The rod is assumed to be thermally isolated everywhere. Starting with a given temperature distribution $u(x, 0)$ over the rod, we are interested in the time evolution of the temperature profile $u(x, t)$. As will be explained in Example 2.3d in Chapter 2, the speed of the heat conduction along the rod is determined by the so-called thermal conductivity $\kappa$. The dimension of $\kappa$ is $\frac{L^2}{T}$ . As an initial profile we take a distribution which is everywhere vanishing except for a peak in the origin: $$ u(x,0) = u_0\delta(x), $$ where $\delta(x)$ is the so-called delta function.
a) Determine the dimensions of the constant $u_0$. To that end, integrate the initial condition over some interval containing $x=0$ and use the properties of the delta function.
I'm at a loss how to solve this. I know that $u$ is a temperature and its dimension is $\frac{ML^2}{T^2}$. However, what happens with its dimension when I integrate? It seems to me that I should integrate over $x=0$ to $x=l$, but I do not know how to incorporate the properties of the delta function. I do know that if we integrate over a domain $I$ containing zero, it follows that $$\int_I \delta(x)v(x)\,dx = v(0),$$ for a continuous function $v(x)$. I would appreciate it if someone could give me a hint, since I cannot continue with the rest of the exercise without solving this part first.