Line Integral of a Field

Question

I have solved the attached problem. I wish to find out if my approach to the problem is correct?

Answer 1

You did not multiply the $\frac{128}{7}$ by $2$, but I think everything else is good.

But I would do it this way:
The $\vec{F}$ is: $\vec{F}(\vec{r})=[x^2-y^2, xy]^T$.
Let's use $t$ for the parametrization, so $\vec{r}=[t, t^3]^T$.
Now rewrite $\vec{F}$: $\vec{F}(\vec{r})=[t^2-t^6, t^4]^T$.
Now rewrite $\mathrm{d}\vec{r}$: $\mathrm{d}\vec{r}=\frac{\partial \vec{r}}{\partial t} \mathrm{d}t=[1, 3t^2]^T\mathrm{d}t$
So our integral is: $$\int_C \vec{F}\cdot\mathrm{d}\vec{r}=\int_0^2\vec{F}\cdot\frac{\partial \vec{r}}{\partial t} \mathrm{d}t$$ $$=\int_0^2 t^2-t^6+3t^6\mathrm{d}t$$ $$=\int_0^2 t^2+2t^6\mathrm{d}t$$ $$=\frac{824}{21}$$