6.2 Line integrals  (Page 11/20)

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True or False? If C is given by $x\left(t\right)=t\text{,}\phantom{\rule{0.2em}{0ex}}y\left(t\right)=t\text{, 0}\le \text{t}\le 1,$ then ${\int }_{C}^{}xyds={\int }_{0}^{1}{t}^{2}dt.$

False

For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path.

[T] ${\int }_{C}^{}\left(x+y\right)ds$

$C\text{:}\phantom{\rule{0.2em}{0ex}}x=t,y=\left(1-t\right)\text{,}\phantom{\rule{0.2em}{0ex}}z=0$ from (0, 1, 0) to (1, 0, 0)

[T] ${\int }_{C}^{}\left(x-y\right)ds$

$C\text{:}\phantom{\rule{0.2em}{0ex}}\text{r}\left(t\right)=4t\text{i}+3t\text{j}$ when $0\le t\le 2$

${\int }_{C}^{}\left(x-y\right)ds=10$

[T] ${\int }_{C}^{}\left({x}^{2}+{y}^{2}+{z}^{2}\right)ds$

$C\text{:}\phantom{\rule{0.2em}{0ex}}\text{r}\left(t\right)=\text{sin}\phantom{\rule{0.2em}{0ex}}t\text{i}+\text{cos}\phantom{\rule{0.2em}{0ex}}t\text{j}+8t\text{k}$ 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 $\left(-2,-1\right)$ to (1, 2).

For the following exercises, find the work done.

Find the work done by vector field $\text{F}\left(x,y,z\right)=x\text{i}+3xy\text{j}-\left(x+z\right)\text{k}$ on a particle moving along a line segment that goes from $\left(1,4,2\right)$ to $\left(0,5,1\right).$

$W=8$

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 $\text{F}\left(x,y,z\right)=-\frac{1}{2}x\text{i}-\frac{1}{2}y\text{j}+\frac{1}{4}\text{k}$ on a particle as it moves along the helix $\text{r}\left(t\right)=\text{cos}\phantom{\rule{0.2em}{0ex}}t\text{i}+\text{sin}\phantom{\rule{0.2em}{0ex}}t\text{j}+t\text{k}$ from point $\left(1,0,0\right)$ to point $\left(-1,0,3\pi \right).$

$W=\frac{3\pi }{4}$

Find the work done by vector field $\text{F}\left(x,y\right)=y\text{i}+2x\text{j}$ in moving an object along path C , which joins points (1, 0) and (0, 1).

Find the work done by force $\text{F}\left(x,y\right)=2y\text{i}+3x\text{j}+\left(x+y\right)\text{k}$ in moving an object along curve $\text{r}\left(t\right)=\text{cos}\left(t\right)\text{i}+\text{sin}\left(t\right)\text{j}+\frac{1}{6}\text{k},$ 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 $\rho \left(x,y\right)={y}^{2}.$

For the following exercises, evaluate the line integrals.

Evaluate ${\int }_{C}\text{F}·d\text{r},$ where $\text{F}\left(x,y\right)=-1\text{j},$ and C is the part of the graph of $y=\frac{1}{2}{x}^{3}-x$ from $\left(2,2\right)$ to $\left(-2,-2\right).$

${\int }_{C}\text{F}·d\text{r}=4$

Evaluate ${\int }_{\gamma }^{}{\left({x}^{2}+{y}^{2}+{z}^{2}\right)}^{-1}ds,$ where $\gamma$ is the helix $x=\text{cos}\phantom{\rule{0.2em}{0ex}}t,y=\text{sin}\phantom{\rule{0.2em}{0ex}}t,z=t\left(0\le t\le T\right).$

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 $\left(1,1,1\right)$ to $\left(3,2,0\right).$

${\int }_{C}^{}yzdx+xzdy+xydz=-1$

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).

${\int }_{C}^{}\left({y}^{2}\right)dx+\left(x\right)dy=\frac{245}{6}$

[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.

$\text{F}\left(x,y\right)=\text{−}x\text{i}-2y\text{j}$

$C\text{:}\phantom{\rule{0.2em}{0ex}}y={x}^{3}\phantom{\rule{0.2em}{0ex}}\text{from (0, 0) to (2, 8)}$

$W=-66$

$\text{F}\left(x\text{,}\phantom{\rule{0.2em}{0ex}}y\right)=2xi+y\text{j}$

C : counterclockwise around the triangle with vertices (0, 0), (1, 0), and (1, 1)

$\text{F}\left(x\text{,}\phantom{\rule{0.2em}{0ex}}y\text{,}\phantom{\rule{0.2em}{0ex}}z\right)=x\text{i}+y\text{j}-5z\text{k}$

$\mathit{\text{C}}\text{:}\phantom{\rule{0.2em}{0ex}}\text{r}\left(t\right)=2\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}t\text{i}+2\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}t\text{j}+t\text{k}\text{,}\phantom{\rule{0.2em}{0ex}}0\le t\le 2\pi$

$W=-10{\pi }^{2}$

Let F be vector field $\text{F}\left(x,y\right)=\left({y}^{2}+2x{e}^{y}+1\right)\text{i}+\left(2xy+{x}^{2}{e}^{y}+2y\right)\text{j}.$ Compute the work of integral ${\int }_{C}^{}\text{F}·d\text{r},$ where C is the path $\text{r}\left(t\right)=\text{sin}\phantom{\rule{0.2em}{0ex}}t\text{i}+\text{cos}\phantom{\rule{0.2em}{0ex}}t\text{j}\text{,}\phantom{\rule{0.2em}{0ex}}0\le t\le \frac{\pi }{2}.$

Compute the work done by force $\text{F}\left(x,y,z\right)=2x\text{i}+3y\text{j}-z\text{k}$ along path $\text{r}\left(t\right)=t\text{i}+{t}^{2}\text{j}+{t}^{3}\text{k},$ where $0\le t\le 1.$

$W=2$

Evaluate ${\int }_{C}^{}\text{F}·d\text{r},$ where $\text{F}\left(x,y\right)=\frac{1}{x+y}\text{i}+\frac{1}{x+y}\text{j}$ and C is the segment of the unit circle going counterclockwise from $\left(1,0\right)$ to (0, 1).

Force $\text{F}\left(x,y,z\right)=zy\text{i}+x\text{j}+{z}^{2}x\text{k}$ acts on a particle that travels from the origin to point (1, 2, 3). Calculate the work done if the particle travels:

1. along the path $\left(0,0,0\right)\to \left(1,0,0\right)\to \left(1,2,0\right)\to \left(1,2,3\right)$ along straight-line segments joining each pair of endpoints;
2. along the straight line joining the initial and final points.
3. Is the work the same along the two paths? a. $W=11;$ b. $W=11;$ c. Yes

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