Integrating to an absolute value?

[Blastrix91]

Homework Statement

I need some help understanding an integral step in the example below. I get how the integrand was set up, but I don't get how comes to two expressions with the absolute value of z-L/2.

(Problem description if that is needed: L is the length of a cylinder with radius R, and P is the polarization of the cylinder in direction of its length. Calculate the electric field at a point on the axis of the rod)

Homework Equations

http://imageshack.us/a/img594/1895/absolutevalue.png

The Attempt at a Solution

I don't understand that step really. Google has no answers. I thought that it might have been some absolute value identities, but it seems that was not the case. Is there someone who'd lend a hand?

 

[Ray Vickson]

Homework Statement

I need some help understanding an integral step in the example below. I get how the integrand was set up, but I don't get how comes to two expressions with the absolute value of z-L/2.

(Problem description if that is needed: L is the length of a cylinder with radius R, and P is the polarization of the cylinder in direction of its length. Calculate the electric field at a point on the axis of the rod)

Homework Equations

http://imageshack.us/a/img594/1895/absolutevalue.png

The Attempt at a Solution

I don't understand that step really. Google has no answers. I thought that it might have been some absolute value identities, but it seems that was not the case. Is there someone who'd lend a hand?

$$\int_0^R \frac{r}{\sqrt{r^2+a^2}} dr = \frac{1}{2} \int_{r=0}^R \frac{d(r^2)}{\sqrt{r^2+a^2}} = \frac{1}{2} \int_0^{R^2} \frac{dy}{\sqrt{y+a^2}} = \left. \sqrt{y+a^2}\right|_0^{R^2}$$ Look at what you get if a > 0 or a < 0.

RGV

 

[SammyS]

Homework Statement

I need some help understanding an integral step in the example below. I get how the integrand was set up, but I don't get how comes to two expressions with the absolute value of z-L/2.

(Problem description if that is needed: L is the length of a cylinder with radius R, and P is the polarization of the cylinder in direction of its length. Calculate the electric field at a point on the axis of the rod)

Homework Equations

http://imageshack.us/a/img594/1895/absolutevalue.png

The Attempt at a Solution

I don't understand that step really. Google has no answers. I thought that it might have been some absolute value identities, but it seems that was not the case. Is there someone who'd lend a hand?

It's because, $\displaystyle \sqrt{u^2}=|u|\ .$

 

[Blastrix91]

Wow thank you. That was pretty helpful