Could use some help in evaluating this line integral....
$\displaystyle
\int\limits_\Upsilon {\{ [\exp (x^2 /y^2 ) + xy^2 ]dx + [(y^6 /x) + (x/y^3 )]dy\} }
$
where the path r is the line $\displaystyle {y^2 }$ = x, from (1,1) to (4,2)
First you need to parameterize the curve
let $\displaystyle y=t$ then $\displaystyle x=t^2$ $\displaystyle t \in [1,2]$
Note: there are many other ways we have parameterized this.
Now we can compute dy and dx
$\displaystyle dx=2tdt \mbox{ and } dy=dt$
now we just substitute in to the above and evalute
$\displaystyle \int_{1}^{2}\left( e^{t^2}+t^4\right)(2t)dt+\left( t^4+\frac{1}{t}\right)dt$
Now it is off to the races.
Good luck.