## Mohsin007 3 years ago * The average value of an even function is (-ve ,+ve, infinite, none)

1. vf321

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

2. vf321

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

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

4. Mohsin007

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

5. Mohsin007

6. vf321

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

7. Mohsin007

so the best ans is none...??

8. vf321

yes, I guess.

9. vf321

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$