How Do You Calculate the Mean Radius of a Cylinder?

[kmoh111]

Homework Statement

I am working a homework problem that is trying to find the mean radius, $$\bar{r}$$ from the midpoint of a cylinder.

The problem states:
What is the mean radius, $$\bar{r}$$ from the midpoint of a cylinder of radius a and height h to its boundary surface? Evalute mean radius $$\bar{r}$$ for a = h/2 = 10 cm.

Homework Equations

The relavent equation is $$\bar{r}$$ = (1/4pi)$$\int$$$$\int^$$ r sin$$\vartheta$$d$$\vartheta$$d$$\beta$$

The Attempt at a Solution

The problem and formula above is from Attix's textbook. In this case I think the limits for beta need to be 0 to 2pi for a cylinder.

I'm not sure what the limits for theta should be. I'm think it's 0 to pi.
I need to express r in terms of theta - but I'm not sure how.

Attix gives the answer as 11.32 cm.

Thanks in advance for any assistance.

 

[gabbagabbahey]

I would use cylindrical coordinates $(s,\phi,z)$ if I were you...the distance from the center of the cylinder (the origin) to a general point on the cylinder $(s,\phi,z)$ is then $\sqrt{s^2+z^}$...then all you need to do is average that over all three surfaces of the cylinder.

What is the general formula for averaging a function over a surface $\mathcal{S}$?...Use that.