Standard errors in surface areas and volumes?

[zejaa7]

I have to finish this one question that I have come across and I am having a bit of trouble figuring out where to go from what I havee done.
The Q is:

A copper cylinder is 5.82 +/- 0.06 cm long and has a radius of 2.53 +/- 0.04 cm. Using the appropraite formula,
Question Details
a) Find the total surface area of the cylinder and the standard error in the area .
b) Find the volume of the cylinder and the standard error in the volume.
c) Given that the density of copper is 8.96cm^-3, find the mass of the cylinder

We are to use error propagation formulas for this.

For part a and b I calculated the normals and for b- volume and used the formulae for product rules to calculate the error in volume and found it to be 2.88.

Can someone guide me through the process because I have the formulas and I plug in the values but I am not sure If i am right or not.

Thank you.

 

[CompuChip]

Technically, the correct formula from error analysis is

$$\delta f(x_1, \cdots, x_n) = \sqrt{ \sum_{i = 1}^n \left( \frac{\partial f(x_1, \cdots, x_n)}{\partial x_i} \delta x_i \right)^2 }$$
where $\delta x_i$ is the uncertainty in x_i.

I prefer to remember just this formula and see how it applies in any particular problem. In this case, for example, the volume of the cilinder would be given by $V = \pi r^2 \ell$ where r is the radius and $\ell$ its length.
Since both of these have an uncertainty associated to them, you can calculate
$$\frac{\partial V}{\partial \ell} = \pi r^2, \quad \frac{\partial V}{\partial r} = 2 \pi r \ell$$
and simply plug in your values for $r, \ell, \delta r \text{ and } \delta\ell$.