Help me sort this out

NOTE: THIS IS NOT A HW PROBLEM
While calculating the moment of inertia of a cone, I made a mistake and got confused. Although I juggled the numbers and got the right answer, I have a doubt.

Say I have to calculate the centroid of a cone z² = r².

0 < z < h

It is quite obvious that the x and y co-ordinates of the centroid are zero. As for the z co-ordinate it can be calculated as follows:

[tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] r² dV =

[tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] r² r dr d[tex]\Theta[/tex] dz =

[tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] r³ dr d[tex]\Theta[/tex] dz = [tex]\Pi[/tex]h⁵/10

But instead, in step 2 if I substituted r = z (from the equation of the cone), I get a completely different answer.

[tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] z² z dr d[tex]\Theta[/tex] dz =

[tex]\int[/tex] [tex]\int[/tex] [tex]\int[/tex][tex]_{V}[/tex] z³ dr d[tex]\Theta[/tex] dz = 2[tex]\Pi[/tex]h⁵/5

I got a similar doubt while calculating the center of mass, wherein I plugged in z = r in the equation and got a different (and wrong) answer.

Shouldn't I get the same answer either way? Why don't I get it?

Anirudh

 

:rofl:

For the first one I used:

[tex]\int[/tex][tex]^{2\Pi}_{\Theta = 0}[/tex][tex]\int[/tex][tex]^{h}_{z = 0}[/tex][tex]\int[/tex][tex]^{z}_{r = 0}[/tex] r³ dr d[tex]\Theta[/tex] dz

For the second one I used:


[tex]\int[/tex][tex]^{2\Pi}_{\Theta = 0}[/tex][tex]\int[/tex][tex]^{h}_{z = 0}[/tex][tex]\int[/tex][tex]^{z}_{z = 0}[/tex] z³ dr d[tex]\Theta[/tex] dz

I used the same limits because r = z. Is that wrong? (Obviously, it MUST be as the answer isn't right)

 

Okay I understood that r = z only on the boundary. What does the above statement mean though?