Question

The density ρ of a certain cylindrical glass rod that has radius 0.05 metres
and length 3 metres varies along the length x of the rod by the formula
ρ(x) = 2800 −
√
32000x kilograms per cubic metre for 0 ≤ x ≤ 3. Find
the total mass of the rod.

i just don't know were to begin in this problem

Answer 1

Anytime you have problem like this or something similar to this where some of the variable is not constant and is changing as you change some other variable ( in this case ro (density) is changing as x is changing) you need to use integrals to solve problem.

First (always do like this) you write what is the mass of the small part of rod that has length of dx (dx represents elementar length). You know that mass = density* volume   which means we need to find elementar volume

Elementar volume for this rod is:
R^2*pi*dx         (R^2*pi is surface of cross section of rod and when we multiply it with dx we get elementar volume - elementar cilinder)

dm (elementar mass) = ro*R^2*pi*dx

Now we have our elementar equation. All we need to do now is to integral both sides

$\int\limits^m_0 {} \, dm = \int\limits^3_0 {R^2*pi*ro} \, *dx$

All you need to do now is to solve integral. (remember to express ro before solving right side, left side is simply  equal to m).