Examples

Line integral in the plane

Line integral in the plane

Problem: Compute the line integral:

$${\int }_{C}ydx+zdy+xdz$$

where $C$ is the line segment with endpoints $(2,0,0)$ and $(3,4,5)$.
Solution: We use the formula $\mathbf{r}(t)=(1-t){\mathbf{r}}_0+t{\mathbf{r}}_1$ for a line passing through ${\mathbf{r}}_0$ and ${\mathbf{r}}_1$. This gives $\mathbf{r}(t)=(2+t,4t,5t)$. We also have $dx=dt$, $dy=4dt$, $dz=5dt$. Plugging everything in:

$${\int }_{C}ydx+zdy+xdz={\int }_0^{1}4t+5t\cdot 4+(2+t)\cdot 5dt={\int }_0^{1}10+29tdt=\mathrm{24.5.}$$

Total mass of parabolic cable

Total mass of parabolic cable

Problem: A cable lies in ${\mathbb{R}}^2$ in the shape of a parabola satisfying $y=x^2$. The density of the cable is given by $\rho (x,y)=y/x$. Compute the total mass of the cable between $x=0$ and $x=4$.
Solution: We use the parametrization of the parabola $P$ given by $\mathbf{r}(t)=(t,t^2)$ with $0\le t\le 4$. The total mass $M$ is given by the integral of density over the arc length of the curve:

$$M={\int }_P\rho (x,y)ds={\int }_0^4\rho (t,t^2)\cdot |(1,2t)|dt={\int }_0^{4}t\sqrt{1+4t^2}dt\approx \mathrm{43.6.}$$