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True or False? If C is given by $x(t)=t\text{,}\phantom{\rule{0.2em}{0ex}}y(t)=t\text{, 0}\le \text{t}\le 1,$ then ${\int}_{C}^{}xyds={\displaystyle {\int}_{0}^{1}{t}^{2}dt}}.$
False
For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path.
[T] ${\int}_{C}^{}(x+y)ds$
$C\text{:}\phantom{\rule{0.2em}{0ex}}x=t,y=(1-t)\text{,}\phantom{\rule{0.2em}{0ex}}z=0$ from (0, 1, 0) to (1, 0, 0)
[T] ${\int}_{C}^{}(x-y)ds$
$C\text{:}\phantom{\rule{0.2em}{0ex}}\text{r}(t)=4t\text{i}+3t\text{j}$ when $0\le t\le 2$
${\int}_{C}^{}(x-y)ds}=10$
[T] ${\int}_{C}^{}({x}^{2}+{y}^{2}+{z}^{2})ds$
$C\text{:}\phantom{\rule{0.2em}{0ex}}\text{r}(t)=\text{sin}\phantom{\rule{0.2em}{0ex}}t\text{i}+\text{cos}\phantom{\rule{0.2em}{0ex}}t\text{j}+8t\text{k}$ when $0\le t\le \frac{\pi}{2}$
[T] Evaluate ${\int}_{C}^{}x{y}^{4}ds},$ where C is the right half of circle ${x}^{2}+{y}^{2}=16$ and is traversed in the clockwise direction.
${\int}_{C}^{}x{y}^{4}ds}=\frac{8192}{5$
[T] Evaluate ${\int}_{C}^{}4{x}^{3}ds},$ where C is the line segment from $(\mathrm{-2},\mathrm{-1})$ to (1, 2).
For the following exercises, find the work done.
Find the work done by vector field $\text{F}(x,y,z)=x\text{i}+3xy\text{j}-(x+z)\text{k}$ on a particle moving along a line segment that goes from $(1,4,2)$ to $(0,5,1).$
$W=8$
Find the work done by a person weighing 150 lb walking exactly one revolution up a circular, spiral staircase of radius 3 ft if the person rises 10 ft.
Find the work done by force field $\text{F}(x,y,z)=-\frac{1}{2}x\text{i}-\frac{1}{2}y\text{j}+\frac{1}{4}\text{k}$ on a particle as it moves along the helix $\text{r}(t)=\text{cos}\phantom{\rule{0.2em}{0ex}}t\text{i}+\text{sin}\phantom{\rule{0.2em}{0ex}}t\text{j}+t\text{k}$ from point $(1,0,0)$ to point $\left(\mathrm{-1},0,3\pi \right).$
$W=\frac{3\pi}{4}$
Find the work done by vector field $\text{F}(x,y)=y\text{i}+2x\text{j}$ in moving an object along path C , which joins points (1, 0) and (0, 1).
Find the work done by force $\text{F}(x,y)=2y\text{i}+3x\text{j}+(x+y)\text{k}$ in moving an object along curve $\text{r}(t)=\text{cos}(t)\text{i}+\text{sin}(t)\text{j}+\frac{1}{6}\text{k},$ where $0\le t\le 2\pi .$
$W=\pi $
Find the mass of a wire in the shape of a circle of radius 2 centered at (3, 4) with linear mass density $\rho (x,y)={y}^{2}.$
For the following exercises, evaluate the line integrals.
Evaluate ${\int}_{C}\text{F}\xb7d\text{r}},$ where $\text{F}(x,y)=\mathrm{-1}\text{j},$ and C is the part of the graph of $y=\frac{1}{2}{x}^{3}-x$ from $(2,2)$ to $(\mathrm{-2},\mathrm{-2}).$
${\int}_{C}\text{F}\xb7d\text{r}}=4$
Evaluate ${\int}_{\gamma}^{}{\left({x}^{2}+{y}^{2}+{z}^{2}\right)}^{\mathrm{-1}}ds},$ where $\gamma $ is the helix $x=\text{cos}\phantom{\rule{0.2em}{0ex}}t,y=\text{sin}\phantom{\rule{0.2em}{0ex}}t,z=t(0\le t\le T).$
Evaluate ${\int}_{C}^{}yz\phantom{\rule{0.1em}{0ex}}dx+xz\phantom{\rule{0.1em}{0ex}}dy+xy\phantom{\rule{0.1em}{0ex}}dz$ over the line segment from $(1,1,1)$ to $(3,2,0).$
${\int}_{C}^{}yzdx+xzdy+xydz}=\mathrm{-1$
Let C be the line segment from point (0, 1, 1) to point (2, 2, 3). Evaluate line integral ${\int}_{C}^{}yds}.$
[T] Use a computer algebra system to evaluate the line integral ${\int}_{C}{y}^{2}dx+xdy},$ where C is the arc of the parabola $x=4-{y}^{2}$ from (−5, −3) to (0, 2).
${\int}_{C}^{}\left({y}^{2}\right)dx+(x)dy}=\frac{245}{6$
[T] Use a computer algebra system to evaluate the line integral ${\int}_{C}^{}\left(x+3{y}^{2}\right)dy$ over the path C given by $x=2t\text{,}\phantom{\rule{0.2em}{0ex}}y=10t\text{,}$ where $0\le t\le 1.$
[T] Use a CAS to evaluate line integral ${\int}_{C}^{}xydx+ydy$ over path C given by $x=2t\text{,}\phantom{\rule{0.2em}{0ex}}y=10t\text{,}$ where $0\le t\le 1.$
$\int}_{C}^{}xydx+ydy=\frac{190}{3$
Evaluate line integral ${\int}_{C}^{}\left(2x-y\right)dx+\left(x+3y\right)dy},$ where C lies along the x -axis from $x=0\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}x=5.$
[T] Use a CAS to evaluate ${\int}_{C}^{}\frac{y}{2{x}^{2}-{y}^{2}}ds},$ where C is $x=t\text{,}\phantom{\rule{0.2em}{0ex}}y=t\text{,}\phantom{\rule{0.2em}{0ex}}1\le t\le 5.$
${\int}_{C}\frac{y}{2{x}^{2}-{y}^{2}}ds=\sqrt{2}\phantom{\rule{0.2em}{0ex}}\text{ln}\phantom{\rule{0.2em}{0ex}}5$
[T] Use a CAS to evaluate ${\int}_{C}xyds,$ where C is $x={t}^{2},y=4t,0\le t\le 1.$
In the following exercises, find the work done by force field F on an object moving along the indicated path.
$\text{F}(x,y)=\text{\u2212}x\text{i}-2y\text{j}$
$C\text{:}\phantom{\rule{0.2em}{0ex}}y={x}^{3}\phantom{\rule{0.2em}{0ex}}\text{from (0, 0) to (2, 8)}$
$W=\mathrm{-66}$
$\text{F}(x\text{,}\phantom{\rule{0.2em}{0ex}}y)=2xi+y\text{j}$
C : counterclockwise around the triangle with vertices (0, 0), (1, 0), and (1, 1)
$\text{F}(x\text{,}\phantom{\rule{0.2em}{0ex}}y\text{,}\phantom{\rule{0.2em}{0ex}}z)=x\text{i}+y\text{j}-5z\text{k}$
$\mathit{\text{C}}\text{:}\phantom{\rule{0.2em}{0ex}}\text{r}(t)=2\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}t\text{i}+2\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}t\text{j}+t\text{k}\text{,}\phantom{\rule{0.2em}{0ex}}0\le t\le 2\pi $
$W=\mathrm{-10}{\pi}^{2}$
Let F be vector field $\text{F}(x,y)=\left({y}^{2}+2x{e}^{y}+1\right)\text{i}+\left(2xy+{x}^{2}{e}^{y}+2y\right)\text{j}.$ Compute the work of integral ${\int}_{C}^{}\text{F}\xb7d\text{r}},$ where C is the path $\text{r}(t)=\text{sin}\phantom{\rule{0.2em}{0ex}}t\text{i}+\text{cos}\phantom{\rule{0.2em}{0ex}}t\text{j}\text{,}\phantom{\rule{0.2em}{0ex}}0\le t\le \frac{\pi}{2}.$
Compute the work done by force $\text{F}(x,y,z)=2x\text{i}+3y\text{j}-z\text{k}$ along path $\text{r}(t)=t\text{i}+{t}^{2}\text{j}+{t}^{3}\text{k},$ where $0\le t\le 1.$
$W=2$
Evaluate ${\int}_{C}^{}\text{F}\xb7d\text{r}},$ where $\text{F}(x,y)=\frac{1}{x+y}\text{i}+\frac{1}{x+y}\text{j}$ and C is the segment of the unit circle going counterclockwise from $(1,0)$ to (0, 1).
Force $\text{F}(x,y,z)=zy\text{i}+x\text{j}+{z}^{2}x\text{k}$ acts on a particle that travels from the origin to point (1, 2, 3). Calculate the work done if the particle travels:
a. $W=11;$ b. $W=11;$ c. Yes
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