Integral of the following Dirac delta distribution

Question

$$\int\limits_0^lC\delta(x-x_0)dx=?$$ where $l, C$ and $x_0$ are constants.

Is the result of the integral $C$?

Answer 1

It depends. If $x_0 \in (0,l)$, then yes, the integral gives C.

Can you think of the limit cases in which $x_0=0$ or $x_0=l$? Recall the definition of the Dirac delta function.

Answer 2

The answer is $C$ if $0<x_0<l$. If $x_0>l$ or $x_0<0$ the answer is 0.

Answer 3

Further to the previous answers, we can write the result as $C(\theta(l-x_0)-\theta(-x_0))$ with $\theta$ the Heaviside step function.