6.2 Line integrals  (Page 11/20)

True or False? If C is given by $x(t)=t\text{,}y(t)=t\text{, 0}\le \text{t}\le 1,$ then ${\displaystyle {\int }_C^{}xyds={\displaystyle {\int }_0^1{t}^{2}dt}}.$

[T] ${\displaystyle {\int }_C^{}(x+y)ds}$

$C\text{:}x=t,y=(1-t)\text{,}z=0$ from (0, 1, 0) to (1, 0, 0)

[T] ${\displaystyle {\int }_C^{}(x-y)ds}$

$C\text{:}\text{r}(t)=4t\text{i}+3t\text{j}$ when $0\le t\le 2$

[T] ${\displaystyle {\int }_C^{}(x^2+y^2+z^2)ds}$

$C\text{:}\text{r}(t)=\text{sin}t\text{i}+\text{cos}t\text{j}+8t\text{k}$ when $0\le t\le \frac{\pi }{2}$

[T] Evaluate ${\displaystyle {\int }_C^{}x{y}^{4}ds},$ where C is the right half of circle $x^2+y^2=16$ and is traversed in the clockwise direction.

[T] Evaluate ${\displaystyle {\int }_C^{}4x^{3}ds},$ where C is the line segment from $(\mathrm{-2},\mathrm{-1})$ to (1, 2).

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 $(0,5,1).$

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 $\text{r}(t)=\text{cos}t\text{i}+\text{sin}t\text{j}+t\text{k}$ from point $(1,0,0)$ to point $\left(\mathrm{-1},0,3\pi \right).$

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 $0\le t\le 2\pi .$

Find the mass of a wire in the shape of a circle of radius 2 centered at (3, 4) with linear mass density $\rho (x,y)=y^2.$

Evaluate ${\displaystyle {\int }_C\text{F}·d\text{r}},$ 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 $(\mathrm{-2},\mathrm{-2}).$

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

Evaluate ${\displaystyle {\int }_C^{}yzdx+xzdy+xydz}$ over the line segment from $(1,1,1)$ to $(3,2,0).$

Let C be the line segment from point (0, 1, 1) to point (2, 2, 3). Evaluate line integral ${\displaystyle {\int }_C^{}yds}.$

[T] Use a computer algebra system to evaluate the line integral ${\displaystyle {\int }_C{y}^{2}dx+xdy},$ where C is the arc of the parabola $x=4-y^2$ from (−5, −3) to (0, 2).

[T] Use a computer algebra system to evaluate the line integral ${\displaystyle {\int }_C^{}\left(x+3y^2\right)dy}$ over the path C given by $x=2t\text{,}y=10t\text{,}$ where $0\le t\le 1.$

[T] Use a CAS to evaluate line integral ${\displaystyle {\int }_C^{}xydx+ydy}$ over path C given by $x=2t\text{,}y=10t\text{,}$ where $0\le t\le 1.$

Evaluate line integral ${\displaystyle {\int }_C^{}\left(2x-y\right)dx+\left(x+3y\right)dy},$ where C lies along the x -axis from $x=0\text{to}x=5.$

[T] Use a CAS to evaluate ${\displaystyle {\int }_C^{}\frac{y}{2x^2-y^2}ds},$ where C is $x=t\text{,}y=t\text{,}1\le t\le 5.$

[T] Use a CAS to evaluate ${\displaystyle {\int }_{C}xyds,}$ where C is $x=t^2,y=4t,0\le t\le 1.$

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

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

$\text{F}(x\text{,}y)=2xi+y\text{j}$

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

$\text{F}(x\text{,}y\text{,}z)=x\text{i}+y\text{j}-5z\text{k}$

$\mathit{\text{C}}\text{:}\text{r}(t)=2\text{cos}t\text{i}+2\text{sin}t\text{j}+t\text{k}\text{,}0\le t\le 2\pi$

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 ${\displaystyle {\int }_C^{}\text{F}·d\text{r}},$ where C is the path $\text{r}(t)=\text{sin}t\text{i}+\text{cos}t\text{j}\text{,}0\le t\le \frac{\pi }{2}.$

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 $0\le t\le 1.$

Evaluate ${\displaystyle {\int }_C^{}\text{F}·d\text{r}},$ 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:

1. along the path $(0,0,0)\to (1,0,0)\to (1,2,0)\to (1,2,3)$ 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?