
Evaluate the line integral ${\int}_{C}y\phantom{\rule{0.2em}{0ex}}ds$,
where $C$ is the curve $x={t}^{2}$, $y=t$,
$0\le t\le 2$.
$\frac{17\sqrt{17}1}{12}$

Evaluate the line integral
${\int}_{C}x{y}^{4}\phantom{\rule{0.2em}{0ex}}ds$,
where $C$ is the right half of the circle ${x}^{2}+{y}^{2}=16$.
$1638.4$

Evaluate the line integral ${\int}_{C}x{y}^{3}\phantom{\rule{0.2em}{0ex}}ds$,
where $C$ is the curve $x=4\mathrm{sin}\left(t\right)$,
$y=4\mathrm{cos}\left(t\right)$,
$z=3t$,
$0\le t\le \pi /2$.
$320$

Evaluate the line integral
${\int}_{C}x{e}^{yz}\phantom{\rule{0.2em}{0ex}}ds$,
where $C$ is the line segment from $\left(0,0,0\right)$
to $\left(1,2,3\right)$.
$\frac{\sqrt{14}\left({e}^{6}1\right)}{12}$

Evaluate the line integral
${\int}_{C}xy\phantom{\rule{0.2em}{0ex}}dx+\left(xy\right)\phantom{\rule{0.2em}{0ex}}dy$
where $C$ consists of line segments from $\left(0,0\right)$
to $\left(2,0\right)$
and from $\left(2,0\right)$
to $\left(3,2\right)$.
$\frac{17}{3}$

Evaluate the line integral
${\int}_{C}\mathrm{sin}\left(x\right)\phantom{\rule{0.2em}{0ex}}dx+\mathrm{cos}\left(y\right)\phantom{\rule{0.2em}{0ex}}dy$
where $C$ consists of the top half of the unit circle
from $\left(1,0\right)$
to $\left(1,0\right)$
and the line
segment from $\left(1,0\right)$
to $\left(2,3\right)$.

Evaluate the line integral
${\int}_{C}{x}^{2}\phantom{\rule{0.2em}{0ex}}dx+{y}^{2}\phantom{\rule{0.2em}{0ex}}dy+{z}^{2}\phantom{\rule{0.2em}{0ex}}dz$,
where $C$ consists of line segments
from $\left(0,0,0\right)$
to $\left(1,2,1\right)$,
and from $\left(1,2,1\right)$
to $\left(3,2,0\right)$.

Evaluate the line integral
$\int}_{C}\mathbf{F}\u2022\phantom{\rule{0.2em}{0ex}}d\mathbf{r$
where $\mathbf{F}\left(x,y\right)={x}^{2}{y}^{3}\mathbf{i}y\sqrt{x}\mathbf{j}$
and $C$ is given by the vector function
$\mathbf{r}\left(t\right)={t}^{2}\mathbf{i}{t}^{3}\mathbf{j}$,
$0\le t\le 1$.
$\frac{59}{105}$

Evaluate the line integral
$\int}_{C}\mathbf{F}\u2022\phantom{\rule{0.2em}{0ex}}d\mathbf{r$
where $\mathbf{F}\left(x,y,z\right)=\u27e8z,y,x\u27e9$
and $C$ is given by the vector function
$\mathbf{r}\left(t\right)=\u27e8t,\mathrm{sin}\left(t\right),\mathrm{cos}\left(t\right)\u27e9$,
$0\le t\le \pi $.

A thin wire is bent into the shape of a semicircle
${x}^{2}+{y}^{2}=4$,
$x\ge 0$.
If the linear density is a constant $k$,
find the mass and the centroid of the wire.
$2\pi k$ and $\left(4/\pi ,0\right)$

Find the work done by the force field
$\mathbf{F}\left(x,y,z\right)=\u27e8y+z,x+z,x+y\u27e9$
on a particle that moves along the line segment
from $\left(1,0,0\right)$
to $\left(3,4,2\right)$.
$26$

 An intelligent robot ($160$ kg) is carrying a canister
with oil ($25$ kg) up a helical ramp that encircles a
rocket with a radius $20$ m. If the rocket is $90$ m high
and the robot makes exactly $3$ complete revolutions, how
much work is done by the robot against gravity while
climbing to the top?

If there is a hole in the canister and $9$ kg of oil
leaks out at a steady rate during the ascent, how much
work is done?

An object moves along the curve $C$ shown
in the figure from $\left(1,2\right)$ to $\left(9,8\right)$.
The lengths of the vectors in the force field
$\mathbf{F}$ are measured in newtons by the scales on the axes.
Estimate the work done by $\mathbf{F}$ on the object.
$22$ J