How to find the moment of inertia for a cylindrical segment

I want to find the moment of inertia for a cylindrical segment, show below:

On Wolfram MathWorld, I found a formula for the volume of a cylindrical segment. Let

$$h(r,theta) = h_1 +frac{1}{2}left(1+frac{r}{R}cos(theta) right) (h_2-h_1)$$
Then
$$V =int_0^R int_0^{2pi}int_{0}^{h(r,theta)}rdzdtheta dr$$

Note, I’m using cylindrical coordinates. My source is here

https://mathworld.wolfram.com/CylindricalSegment.html#:~:text=A%20cylindrical%20segment%2C%20sometimes%20also,known%20as%20a%20cylindrical%20wedge.

(Note they have typo in the order of integration)

However, I’ve recently been calculating a lot of moments of inertia and I almost always need to integrate $z$ from $-frac{h}{2}$ to $frac{h}{2}$. So I’m concerned the limits of integration Mathworld suggests for the volume would not work for the moment of inertia.

I think I can show easily this integral is not correct. If this integral were correct, we could set $h_1 = h_2 = h$. When we do this, $h(r,theta) = h$. Then the integral becomes

$$mathbf{I} = int_{0}^{R}int_{0}^{2pi}int_{0}^{h}begin{bmatrix}
y^2 + z^2 & -xy & -xz
-yx & x^2 + z^2 & -yz
-zx & -zy & x^2 + y^2
end{bmatrix} r dzdtheta dr$$

If you do this, you get
$$
left(
begin{array}{ccc}
frac{1}{12} m left(4 h^2+3 R^2right) & 0 & 0
0 & frac{1}{12} m left(4 h^2+3 R^2right) & 0
0 & 0 & frac{m R^2}{2}
end{array}
right)
$$
which does not equal the moment of inertia tensor of a cylinder, which you can find here.

https://en.wikipedia.org/wiki/List_of_moments_of_inertia

However, if you do

$$int_0^R int_0^{2pi}int_{-frac{h(r,theta)}{2}}^{frac{h(r,theta)}{2}}begin{bmatrix}
y^2 + z^2 & -xy & -xz
-yx & x^2 + z^2 & -yz
-zx & -zy & x^2 + y^2
end{bmatrix}rdzdtheta dr$$

Then in the simple case of $h_1 = h_2 = h$, you get

$$left(
begin{array}{ccc}
frac{1}{12} m left(h^2+3 R^2right) & 0 & 0
0 & frac{1}{12} m left(h^2+3 R^2right) & 0
0 & 0 & frac{m R^2}{2}
end{array}
right)$$

which is the correct moment of inertia tensor for a regular cylinder.

However, I’m not sure if this special case is sufficient to show that it works in general for a cylindrical segment. I only showed it is true for a regular cylinder.

Can someone explain intuitively why these limits of integration would be appropriate for the case of the cylindrical segment? I don’t really get intuitively how this integral works geometrically to represent the cylindrical segment, so if someone could explain intuitively why it makes sense (or doesn’t) that would be great. Then I could be confident conceptually it is right. At the moment, all I’m relying on is this one check of reproducing a regular cylinder’s inertia tensor and I have no physics intuition at all.