Integral of de absolute value of a derivative

[kurushishraqi]

Hi!

Homework Statement

I'm trying to work out this integral, without success:

$\int_{-a}^{a} (1-\left|x\right|/a) \frac{d^2}{dx^2}(1-\left|x\right|/a)$

2. The attempt at a solution

I tried solving this by parts, but I'm stuck:

$\int_{-a}^{a} (1-\left|x\right|/a) \frac{d^2}{dx^2}(1-\left|x\right|/a)$
$=(1-\left|x\right|/a) \frac{d}{dx}(1-\left|x\right|/a)|_{-a}^{a}- \int_{-a}^{a} (\frac{d}{dx}(1-\left|x\right|/a))^2$Now, the first term seems to be wrong; the derivative of abs(x) is not defined, and with the second term, more of that. I tried to split the integral in two parts, for positive and negative x's, but that gives me 0. I think that result is wrong, because the function $(1-\left|x\right|/a)$ is concave, and I'm expecting a positive second derivative.

Well, any input would be highly appreciated.

Thanks.

 

[Ray Vickson]

Hi!

Homework Statement

I'm trying to work out this integral, without success:

$\int_{-a}^{a} (1-\left|x\right|/a) \frac{d^2}{dx^2}(1-\left|x\right|/a)$

2. The attempt at a solution

I tried solving this by parts, but I'm stuck:

$\int_{-a}^{a} (1-\left|x\right|/a) \frac{d^2}{dx^2}(1-\left|x\right|/a)$
$=(1-\left|x\right|/a) \frac{d}{dx}(1-\left|x\right|/a)|_{-a}^{a}- \int_{-a}^{a} (\frac{d}{dx}(1-\left|x\right|/a))^2$


Now, the first term seems to be wrong; the derivative of abs(x) is not defined, and with the second term, more of that. I tried to split the integral in two parts, for positive and negative x's, but that gives me 0. I think that result is wrong, because the function $(1-\left|x\right|/a)$ is concave, and I'm expecting a positive second derivative.

Well, any input would be highly appreciated.

Thanks.

(d/dx)[1-|x|/a] is +1/a for x < 0 and is -1/a for x > 0 (and is undefined at x = 0). If you feel confident applying integration by parts in the presence of such discontinuities, you can use the above to complete the calculation. Alternatively, you can regard |x|/a as the limit of sqrt(e+x^2/a^2) as e --> 0 from above, do the integral for finite e > 0, then take the limit in the final result.

RGV

 

[kurushishraqi]

Thanks a lot!

 

[rude man]

Where's the dx in this integral? There is no infinitesimal integrating element in your integral.

 

[kurushishraqi]

It's called a typo.