Hello. I'm having a bit of a problem calculating the 2nd moment of area of 2D shapes etc.

The Question:

Calculate $\displaystyle I_x$ for the section sketched in Figure 1 to within about 2%.

(I have attached the figure as a JPEG - all measurements are in millimetres).

Useful equations

In my notes, there are 3 moments of area defined:

$\displaystyle I_z = \int_A y^2 \,dA $

$\displaystyle I_y = \int_A z^2 \,dA $

$\displaystyle J = I_x = \int_A r^2 \,dA = \int_A (y^2+z^2)dA = I_z+I_y $

Where J is the POLAR 2nd moment of area.

My Instructor's Solution

$\displaystyle I_{zz} = \bigg[ \frac{1}{12} \times 300 \times 25^3 + 300 \times 25 \times \bigg(\frac{500-25}{2}\bigg)^2\bigg] \times 2 + \frac{1}{12} \times 12 \times 450^3 $

$\displaystyle = 9.38 \times 10^8 \, mm^4 $

My Confusion

1) In my instructors solution he calculated $\displaystyle I_{zz} $ which is a notation he has never used in his notes? Does this mean the same as $\displaystyle I_x $, which is what the question asked for? Or has he calculated what he calls $\displaystyle I_z $ in his notes?

2) In his solution he seems to be pulling figures straight out of his arse, and there is no integration involved at all, despite the fact that the definitions of 2nd moments of intertia are all integrals.

Can anyone offer an explanation?