## Mohsin007 Group Title * The average value of an even function is (-ve ,+ve, infinite, none) one year ago one year ago

1. vf321 Group Title

Well that depends really... The average value of its entire domain?

2. vf321 Group Title

I can think of an even function which has a finite average value, and I can think of one that has an infinite one...

3. vf321 Group Title

If you were talking about an odd function then it'd be easy...

4. Mohsin007 Group Title

This was a question in my exam and i dnt knw how to figure it out..

5. Mohsin007 Group Title

6. vf321 Group Title

For an odd function it would be 0, since on the other side of the y axis you have negative values.

7. Mohsin007 Group Title

so the best ans is none...??

8. vf321 Group Title

yes, I guess.

9. vf321 Group Title

But to your previous question: For an odd function $$f$$, we know that $$f(x)=-f(-x)$$. As such, the average value (assuming $$f$$ exists over all $$\mathbb R$$) is$\lim_{a\rightarrow\infty}\frac{1}{2a}\int_{-a}^af(x)dx=\lim_{a\rightarrow\infty}\frac{1}{2a}(\int_0^af(x)dx+\int_{-a}^0f(-x)dx))$$=\lim_{a\rightarrow\infty}\frac{1}{2a}(\int_0^af(x)dx+\int_{-a}^0-f(-x)dx)$Let $$u=-x$$$=\lim_{a\rightarrow\infty}\frac{1}{2a}(\int_0^af(x)dx-\int_{0}^af(u)du)=0$