Scalar Line Integral

${\int }_{\mathcal{C}}f(x,y,z)ds={\int }_{t_0}^{t_f}f(\vec{r}(t))||{\vec{r}}^{\prime }(t)||dt$

• If $f$ represents mass density and $ds$ represents some infinitesimal length of the curve, the scalar line integral would represent the mass of the curve segment
• Might involve finding a parameterization of $f$ in terms of $t$
• Ex: What is the mass of a wire represented by the curve $y=x^3,\{0\le x\le 1\}\text{ (cm)}$ if its mass density is represented by $f(x,y)=\sqrt{1+9xy}\text{ (g/cm)}$
  • $r(t)=⟨t,t^3⟩$
  • $$\begin{gathered} M \\ ={\int }_C\sqrt{1+9xy}ds \\ ={\int }_0^1\sqrt{1+9t^4}||⟨1,3t^2⟩||dt \\ =\frac{14}{5}g \end{gathered}$$