Exercises

01

Line integrals with respect to arc length

Compute the line integrals of the given function over the specified curve:

• (a) $f(x,y,z)=x^2+y^2+z^2$ over the image of $\mathbf{r}(t)=(\cos (t),\sin (t),t)$ for $0\le t\le \pi$
• (b) $f(x,y)=\sqrt{1+9xy}$ over $y=x^3$ for $0\le x\le 1$
• (c) $f(x,y,z)=2x^2+8z$ over $\mathbf{r}(t)=(e^t,t^2,t)$ for $0\le t\le 1$

02

Electric potential from charged cable

The electric potential $V_p$ at a point $p=(a,b,c)$ in space is sourced by a charged circular wire with charge density $\rho (x,y,z)=\left(2-\frac{x}{R}\right)$. The wire $C$ is located at $(R\cos (t),R\sin (t),0)$. The potential is given by the line integral:

$$V_p=k{\int }_C\frac{\rho (x,y,z)}{d_p}ds$$

where $k$ is a constant, $d_p$ is the distance from $p$ to $(x,y,z)$, and $C$ is a circle parametrized by $(R\cos (t),R\sin (t),0)$. Find the electric potential at the point $p=(0,0,c)$.

03

Line integrals compressed notation

Compute the following line integrals in the plane:

• (a) ${\int }_{C}ydx+zdy+xdz$ for $\mathbf{c}(t)=(2+t^{-1},t^3,t^2)$, $0\le t\le 1$
• (b) ${\int }_C\frac{-ydx+xdy}{x^2+y^2}$ for the line segment with endpoints $(1,0)$ and $(0,1)$

04

Line integrals in the plane

Compute the following line integral in the plane:

$${\int }_C(x+2y)dx+x^{2}dy$$

where $C$ is the following curve: