Compare work along a curved and straight path

A force field is $\mathbf F=M(x,y)\mathbf i+N(x,y)\mathbf j$. For a path $y=f(x)$ from $x=a$ to $x=b$, the work is

$$W=\int_a^b\left[M(x,y)+N(x,y)\frac{dy}{dx}\right]dx.$$

A field is conservative if the work from one point to another is independent of the path.

(i) Sketch $C:y^2=4(1-x)$, $x\le1$.2 marks

(ii) Find $dy/dx$ in terms of $y$.1 mark

(iii) Points $P,Q$ on $C$ have $x=1,-3$ respectively, with $Q$ below the x-axis. Find line $PQ$.2 marks

For the remaining parts, $\mathbf F=x^2\mathbf i+xy^2\mathbf j$.

(iv) Show that work along $C$ from $Q$ to $P$ is $\int_{-3}^1[x^2+4x(1-x)^{1/2}]dx$, then evaluate it exactly.4 marks

(v) Find the work along line $PQ$ from $Q$ to $P$ to 2 decimal places.2 marks

(vi) Determine whether $\mathbf F$ is conservative, with a reason.1 mark