선적분

True or False? Line integral ${\int }_{C}^{}f(x,y)ds$ is equal to a definite integral if C is a smooth curve defined on $[a,b]$ and if function $f$ is continuous on some region that contains curve C.

True or False? Vector functions ${\text{r}}_{1}=t\text{i}+{t}^{2}\text{j},$ $0\le t\le 1,$ and ${\text{r}}_{2}=(1-t)\text{i}+{(1-t)}^{2}\text{j},$ $0\le t\le 1,$ define the same oriented curve.

True or False? ${\int }_{-C}^{}(Pdx+Qdy)={\int }_{C}^{}(Pdx-Qdy)$

True or False? A piecewise smooth curve C consists of a finite number of smooth curves that are joined together end to end.

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

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

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

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

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

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

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

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

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

$C:\,\text{r}(t)=\sin \,t\text{i}+\cos \,t\text{j}+8t\text{k}$ 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.

[T] Evaluate ${\int }_{C}^{}4{x}^{3}ds,$ where C is the line segment from $(-2,-1)$ 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 $\text{r}(t)=\cos \,t\text{i}+\sin \,t\text{j}+t\text{k}$ from point $(1,0,0)$ to point $(-1,0,3\pi ).$

Find the work done by vector field $\text{F}(x,y)=y\text{i}+2x\text{j}$ in moving an object along path C, the straight line 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 $\text{r}(t)=\cos (t)\text{i}+\sin (t)\text{j}+\frac{1}{6}\text{k},$ 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 ${\int }_{C}\text{F}\cdot d\text{r},$ where $\text{F}(x,y)=-1\text{j},$ and C is the part of the graph of $y=\frac{1}{2}{x}^{3}-x$ from $(2,2)$ to $(-2,-2).$

For the following exercises, evaluate the line integrals.

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

For the following exercises, evaluate the line integrals.

Evaluate ${\int }_{C}^{}yzdx+xzdy+xydz$ 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 ${\int }_{C}^{}yds.$