use Green theorem to evaluate line integral: In  the problem below,  parametrize the plane curves below in such a way that it  traversed only once , its unit normal vector towards the interior of the bounded region it enclose. 

Then use Green theorem to evaluate line integral:

$$ \int_C y^2dx+xydy $$

where $ \ C \ $ is the boundary curve  of the region   lying between $ \ y=0 , \ y=\sqrt x \ $ and $ \ x=9 \ $ 

Answer:

Let $ \ y=t, \ x=t^2 \ , \ 0 \leq t \leq 3 \  $ be the parametrization. 

Then, 

$ F(x,y)=y^2 \hat i+xy \hat j, \\ r(t)=x \hat i+y \hat j=t \hat i+t^2 \hat j \\ \Rightarrow dr=(\hat i+2t \hat j)dt  \ \ and \ \ F(r(t))=t^2 \hat i+t^3 \hat j $ 

Thus,

$ \int_C y^2 dx+xy dy =\int_{0}^3 \ F(r) \cdot dr = \int_0^3 (t^2+2t^4) dt = 9+2 \cdot \frac{3^5}{5}$ 

Am I right ?

Help me
Greens Theorem states that 
$$\int_C y^2 dx + xy dy = \iint_D \frac d {dx} xy - \frac d {dy} y^2$$
where $D$ is the region enclosed by $C$. This can be simplified to 
$$= \iint_D y - 2y = \iint_D -y$$
It seems to me the problem wishes you to use Greens Theorem by evaluating this much simpler integral to get the integral over $C$ 

You seemed to try to evaluate the integral without using Greens Theorem which will get you the right answer too, however there are a couple errors:

You swapped the variables in $r(t)$. It should be that $r(t) = x \hat i + y \hat j = t^2 \hat i + t \hat j$. Thus $dr = 2t \hat i + \hat j$ and what you end up integrating is $\int_0^3 2t^3 + t^3\ dt$

You forgot to integrate over the $x=9$ and $y = 0$ portions of $C$.
Greens Theorem states that 
$$\int_C y^2 dx + xy dy = \iint_D \frac d {dx} xy - \frac d {dy} y^2$$
where $D$ is the region enclosed by $C$. This can be simplified to 
$$= \iint_D y - 2y = \iint_D -y$$
It seems to me the problem wishes you to use Greens Theorem by evaluating this much simpler integral to get the integral over $C$ 

You seemed to try to evaluate the integral without using Greens Theorem which will get you the right answer too, however there are a couple errors:

You swapped the variables in $r(t)$. It should be that $r(t) = x \hat i + y \hat j = t^2 \hat i + t \hat j$. Thus $dr = 2t \hat i + \hat j$ and what you end up integrating is $\int_0^3 2t^3 + t^3\ dt$

You forgot to integrate over the $x=9$ and $y = 0$ portions of $C$.