Line intergral of a parameterized density

Question

Q. Find the mass of a string along the curve

$r = 3t\hat{i} + 3t^2\hat{j}+2t^3\hat{k}; \ \ \ \ 0 \leqslant t \leqslant 1$

given that the density at $r(t)$ is $(1+t)$ grams.

I understand how do to this question if the densitys as given in some as $\rho (x,y,z)=....$ but I'm not sure how to approach the problem when it is already parametrized.

Answer 1

$$dM=\rho ds=\rho(\vec{r(t)})|\vec{r^\prime(t)}|dt\\ \implies M=\int_C\left(1+t\right)|\vec{r^\prime(t)}|dt$$ Here, $\vec{r^\prime(t)}=3\hat{i}+6t\hat{j}+6t^2\hat{k}$.

Answer 2

Hint. Density gives the mass per length, so you integrate $$\text{mass}=\text{density}\cdot\text{length}=(1+t)\Delta r(t)=(1+t)||r'(t)||\,\mathrm{d}t$$ where $0\le t\le 1$.