Need help with finding Center of Gravity with given radius and height

[yang09]

Homework Statement

Caution: Ignore the weight of the bucket
itself.
The bottom and the top of a bucket have
radii rb = 15 cm and rt = 26 cm respectively.
The bucket is h = 36 cm high and filled with
water.
Where is the center of gravity relative to
the center of the bottom of the bucket?
Answer in units of cm.

Homework Equations

Xcm = (X1M1)/(M1)

The Attempt at a Solution

I'm enrolled in a Calculus Physics so I know that I should be using Calculus and not regular physics formula.
Do I have to integrate the problem, but the only problem is that no equation is given. That's where I am stuck at.
I don't want you to tell me how to do it, but can you give me a hint as where I should start off.

 

[Gregg]

The centre of gravity is going to be on the z axis, but you do not know how high above the point (0,0,0) it is. If you take this point to be the centre of the bottom of the bucket. Switch over to cylinderical co-ordinates and find $$\bar{z}\int dM=\int z dM$$

 

[yang09]

I still don't get it. You say integrate the Mass, but you're not given any formula to integrate. You're only give points. If I was given a formula, I know how to integrate but your not given one to integrate. Do you make one up with the points?

 

[ideasrule]

Have you learned how to find the center of mass using integrals? See this: http://en.wikipedia.org/wiki/Center_of_mass

 

[yang09]

I know how to find Center of gravity using integrals only when an equation is given.
I don't know how to find it without an equation.
How would I go about and integrate without an equation?

 

[Gregg]

If you wanted to know the volume you would integrate

$$\int_0^h \pi (R_1+(R_2-R_1)\frac{z}{h})^2 dz$$

if the mass is uniform you can extend this to the formula you were given earlier?

Doing that I get 18.8cm