Moment of inertia about the origin for the lamina

[fonseh]

Homework Statement

find the moment of inertia about the origin for the lamina which the surface of sphere (x^2) + (y^2) +(z^2) = 9 . z>2 . Given that density is a constant . Here's my wroking

The ans is 16pi (k) , but my ans is different , is my ans wrong ?
If so , which part is wrong ?
I have checked thru the working many times, yet , still can't find the error

Homework Equations

The Attempt at a Solution

 

[Jonathan Scott]

Note that a moment of inertia is about an axis, not a point. In this case I assume you meant the z axis.

I haven't worked right through your answer, and it contains some errors, for example the expression for $z$ is missing a square root, which however seems to have been assumed below. You then appear to have done a substitution which doesn't seem necessary and makes it more complicated.

I would do this question by integrating over circular bands of constant $z$, where the mass of each band is $2 \pi r$ times its width which is $\sqrt{dz^2 + dr^2}$ where obviously you need to express $dr$ in terms of $dz$ before you integrate it. The resulting integral is simple. I make the answer $16 \pi k$ where $k$ is the area density.