How to calculate this Delta Dirac Integral?

Question

I'm trying to understand the result of this integral.

$$\int_{-\infty}^a\delta(x-b)\,\mathrm dx$$

The book gives me the following answer.

If $a>b$, $1$; If $a<b$, $0$;

I'm not sure but maybe this relation may be applied.

Box 5.1.3. The Dirac delta function satisfies the following relation $$\int_{-\infty}^a\delta(x-b)\,\mathrm dx=\begin{cases}1&\text{if }a<x_0<b,\\0&\text{otherwise,}\end{cases}\tag{5.7}$$

Equation ($5.4$) is a special case of this, because $-\infty<x_0<+\infty$ for any value of $x_0$.

Answer 1

We have that $$\int_{\alpha}^{\beta} \delta (x-x_0)dx=\left\{\begin{matrix} 1 & \text{ if } \alpha <x_0<\beta\\ 0 & \text{ otherwise } \end{matrix}\right.$$

and you want to calculate the integral $$\int_{-\infty}^a \delta (x-b)dx$$

For $\alpha=-\infty$, $\beta=a$ and $x_0=b$ we have $$\int_{-\infty}^{a} \delta (x-b)dx=\left\{\begin{matrix} 1 & \text{ if } -\infty <b<a\\ 0 & \text{ otherwise } \end{matrix}\right.$$