How Is the Center of Mass Calculated for a Solid Cone?

[Carnivroar]

I am using the textbook called Classical Mechanics by John R. Taylor.

Z = 1/M ∫ ρ z dV = ρ/M ∫ z dx dy dz

On page 89, example 3.2, it says:

"For any given z, the integral over x and y runs over a circle of radius r = Rz / h, giving a factor of πr² = πR²z² / h²."

I wish the book would show the steps. Can someone please help me understand this? I want to know what the limits of integration for dx and dy are.

 

[Carnivroar]

I think I need to relearn the whole concept of integration again... let me give this a try.

At height h the radius of the cone is Rz / h. That is the radius of each "disk" stacked on top of another to form the cone.

The area of that disk with height dz is πR²z² / h² ... the z² stays inside the integral and we get ∫ z³ dz...

But I want to know what the limits for dx and dy were.

 

[Carnivroar]

Okay I figured it out

dx dy => r dr dθ

Limits for dr is 0 up to R... dθ is 0 to 2π