Solving Delta Dirac Integral Homework Statement

[Jillds]

Homework Statement

Solve the integral $\int_0^{3\pi} \delta (sin \theta) d\theta$

Homework Equations

The Attempt at a Solution

I can rewrite $delta (sin \theta)$ as $\sum_{n=-\infty}^{\infty} \frac{\delta(\theta - n\pi)}{|cos (n\pi)|}=\sum_{n=-\infty}^{\infty} \delta(\theta-n\pi)$

So the integral becomes:
$\int_0^{3\pi} \delta (sin \theta) d\theta = \int_0^{3\pi} [\delta (\theta) + \delta (\theta - \pi) + \delta (\theta - 2\pi) + \delta (\theta - 3\pi)] d\theta$

I understand that the dirac-delta function is symmetrical and the integral is taken over the half, so that

$\int_0^{3\pi} \delta (\theta) d\theta = 1/2$

My solution sheet says the others equal to 1, 1, and 1/2 again. But I'm scratching my head how that is. Can someone explain it to me?

 

[Goddar]

Hi. Your interval is [0;3π] so at 0 and 3π the delta function sits on the limit and, as you correctly put for the 0 case, gives you 1/2 (the same convention is sometimes taken for the Heaviside theta). But at π and 2π you are well within the interval so the delta gives 1 as it should...

 

[Jillds]

Thanks very much for the explanation. Yes, I understand.