Changing variables on an integral question

[dmatador]

Say you have an even function f(y) (that is, f(y) = f(-y)) and you want to integrate

$$\int_ \infty^0 yf(y) dy$$

From negative infinity to 0 (sorry, latex wasn't doing what i wanted)

Is it allowed to take the limit to infinity in the positive direction, and negate the y variables within the integral? Or, rather, is there a way to utilize the fact that f(y) is even in order to change variables to end up with
$$-\int_0^\infty yf(y) dy$$

Sorry if this is vague. I'm mostly interested in dealing with the limit. Thank you for any feedback.

 

[arildno]

Yes, you can do that.

This is how the formal substition goes, given that f is an even function:
Set z=-y

Then:
$$\int_{-\infty}^{0}yf(y)dy=\int_{\infty}^{0}(-z)f(-z)(-dz}=\int_{\infty}^{0}zf(z)dz=-\int_{0}^{\infty}zf(z)dz$$

 

[dmatador]

Yes, you can do that.

Thank you very much. I was stuck because I was making these steps based on intuition and couldn't totally convince myself.