Bound surface charge - hollow tube

[Roodles01]

Homework Statement

A thick-walled cylindrical tube of dielectric material has internal radius a/2 and external radius 2a, and its polarization is given in cylindrical coordinates by the expression P = (A/r2) er.

Derive expressions for the net charge on the inner and outer surfaces of a length L of the tube and for the charge within the volume of length L of the tube.

Homework Equations

σ_b = P . n hat
where n hat is the outward-pointing unit vector normal to the surface and P is the polarization.
surface area of cylinder (ignoring ends) = 2∏rL

The Attempt at a Solution

σ_{b total} = σ_{b inner} + σ_{b outer}

σ_{b outer} = (A/(2a)²) * area of cylinder
σ_{b outer} = (A/(2a)²) * 2∏(2a)L
σ_{b outer} = ∏AL/a

Now the bit I can't seem to marry with the model answer . . .

σ_{b inner} = (A/(a/2)²) * 2∏(a)L
σ_{b inner} = (4A/a²) * 2∏(a)L
σ_{b inner} = 8∏AL/a

At least that's how I got it.

The model answer shows it to be
σ_{b inner} = 4∏AL/a

It has to be a problem I'm having with algebra again, but I can't see where.
Could you show me where I went wrong please.

Or am I right?

 

[Roodles01]

Aaaaah!
Details.
Please ignore this.

Yes, the wrong bit is me not looking proplerly.
radius is a/2 not a.
Sorry.

 

[BvU]

Still has me puzzled. In the attachment I read $\vec P = \left ( A/r^3\right ) \, \vec e_r$ (in the text; in the picture it's hard to distinguish if it's a 3 or a 2).

Or is your exercise another one, with the a/2 instead of the a and the 2 instead of the 3 ?