Answer
The mass of the rod is $20~grams$
Work Step by Step
Let $M(x)$ be the mass of the rod of length $x$
$M'(x) = \rho(x) = \frac{1}{\sqrt{x}} = x^{-1/2}$
We can find an expression for $M(x)$:
$M(x) = 2~\sqrt{x}+C$
Since $M(0) = 0,$ then $C = 0$
$M(x) = 2~\sqrt{x}$
We can find $M(100)$:
$M(x) = 2~\sqrt{100} = 20~grams$
The mass of the rod is $20~grams$