Why Does the Integral of a Step Function from -∞ to +∞ Equal Infinity?

[coki2000]

Hello,
I have a question about dirac-delta function. Integral of step function from -infinity to +infinity equals to infinity from the graph of step function.

$$\int_{-\infty}^{\infty}\theta (x) dx=\infty$$

But calculating mathematically

$$\lim_{a\rightarrow\infty}\int_{-a}^{a}\theta(x) dx=\lim_{a\rightarrow\infty}(\delta \left(a \right)-\delta \left(-a\right))=0$$

Because delta function is 0 except x=0. Please explain to me. Thanks

 

[George Jones]

It seems that you're using

$$\theta \left( x \right) = \frac{d \delta}{dx} \left( x \right),$$

when actually (in a distributional sense)

$$\delta \left( x \right) = \frac{d \theta}{dx} \left( x \right).$$

Also, for $a > 0$,

$$\int^a_{-a} \theta \left( x \right) dx = a.$$

 

[coki2000]

It seems that you're using

$$\theta \left( x \right) = \frac{d \delta}{dx} \left( x \right),$$

when actually (in a distributional sense)

$$\delta \left( x \right) = \frac{d \theta}{dx} \left( x \right).$$

Also, for $a > 0$,

$$\int^a_{-a} \theta \left( x \right) dx = a.$$

Oh, yes, thanks for your help.