Originally Posted by
ThePerfectHacker When I see the $\displaystyle ds$ I take that to mean the line intetgral with $\displaystyle \sqrt{[x'(t)]^2+[y'(t)]^2}$
First we have to parametrize the curve,
If you let,
$\displaystyle x=t$ and $\displaystyle y=2\sqrt{t}$
$\displaystyle 1\leq t\leq 9$
Thus,
$\displaystyle \sqrt{[x'(t)]^2+[y'(t)]^2}=\sqrt{1+(3\sqrt{t})^2}=\sqrt{1+9t^2}$
Thus,
$\displaystyle \int_C 2yds=\int_1^9 t\sqrt{1+9t^2}dt$
But, this integral is easy to evaluate, use substitution,
$\displaystyle u=1+9t^2$