Integration problem

[racuna]

The linear density of a rod of length of 1m is given by p(x)=1/sqrt(x), in grams per centimeter, where x is measured in centimeters from one end of the rod. Find the mass of the rod.

I know there has to be an integration somewhere, but that p in the problem was weird looking. Does that weird p mean something else?

P.S. How do you find the mass of a rod?
The text does not offer any examples on this specifi type of problem.

 

[Unco]

Heya, Racuna.

Just a thought (open to critique, of course), but would it suffice to find the mean value of the linear density (which is mass/length) function by
\(\displaystyle \Large \frac{\int^{100}_0 p(x) \, dx}{100 - 0}\)

and multiplying by the length, 100, (so the 100 cancels) to find the mass of the rod?

 

[Gene]

Mass = density times length.
dM = (1/sqrt(x))*dx

 

[racuna]

I just found out that the weird p is the derivative of m, when m is the mass of the rod. That clears a lot up along with what you two told me. Thanks.

 

[soroban]

Hello, racuna!

The linear density of a rod of length of 1m is given by p(x)=1/sqrt(x), in grams per centimeter,
where x is measured in centimeters from one end of the rod.
Find the mass of the rod.

I know there has to be an integration somewhere, but that p in the problem was weird looking.
Does that weird p mean something else?

It's probably a Greek "rho", \(\displaystyle \rho\), usually used to represent density.

P.S. How do you find the mass of a rod?

Are you waiting for some magic formula?
How do we find the mass of anything?
. . We add up the individual masses (it's called integration).

Enough sarcasm . . . the problem is easier than you think.

\(\displaystyle \L M\;=\;\int^{\;\;\;\;100}_0\frac{1}{\sqrt{x}}\,dx\;=\;\int^{\;\;\;\;100}_0x^{-\frac{1}{2}}\,dx\)

Can you finish it now?