$($Chapter $15,16,$ Section $7\mathrm{.}9)$

Evaluate the line integral

${\displaystyle \underset{C}{\overset{\phantom{}}{}}}xyds$

where $C$ is given by the line $y=lnx,1xe.$

$[7]$

$6.($Chapter $15,$ Section $16\mathrm{.}5)$

A particle moves from $(1,1,0)$ to $(-1,1,-1)$ under the influence of the vector field

$F_{x,y}=(2xyz,x^{2}z+2y{z}^2,x^{2}y+2y^{2}z+e^z)$

$($a$)$ Show that the integral is independent of the path followed by using the Fundamental Theorem of Line Integrals $($Theorem $16\mathrm{.}5\mathrm{.}1).$

$($b$)$ Compute the integral

i$.$ by transforming the integral to an ordinary single integral $($see Section $15\mathrm{.}3)$

ii$.$ using the Fundamental Theorem of Line Integrals.

$[11]$