What is the Moment of Inertia of a Cone about its Longitudinal Axis?

[c0der]

Homework Statement

Find the moment of inertia of a solid cone about its longitudinal axis (z-axis)

The cone: x^2+y^2<=z^2, 0<=z<=h

I_z = \int\int\int_T(x^2+y^2)dxdydz

Homework Equations

Representing the cone in cylindrical coords:

x=zcos\theta
y=zsin\theta
z=z

The Attempt at a Solution

The integral in cylindrical coords is:

I_z = \int_0^h \int_0^{2\pi} \int_0^z (z^2)rdrd\theta dz

Evaluating the triple integral gives:
\pi h^5/5

But the answer in the book is:
\pi h^5/10

I don't see what I did wrong

 

[LCKurtz]

Homework Statement

Find the moment of inertia of a solid cone about its longitudinal axis (z-axis)

The cone: x^2+y^2<=z^2, 0<=z<=h

I_z = \int\int\int_T(x^2+y^2)dxdydz

Homework Equations

Representing the cone in cylindrical coords:

x=zcos\theta
y=zsin\theta
z=z

The Attempt at a Solution

The integral in cylindrical coords is:

I_z = \int_0^h \int_0^{2\pi} \int_0^z (z^2)rdrd\theta dz

The moment arm for the second moment from the z axis is $r^2$, not $z^2$.

Evaluating the triple integral gives:
\pi h^5/5

But the answer in the book is:
\pi h^5/10

I don't see what I did wrong

 

[c0der]

I see:

x=rcos\theta, y=rsin\theta where 0<=r<=z

Thanks a lot