Solution (a) Let $P(x,y)=2xy+24x$ and $Q(x,y)=x^{2}+16$. Then $\dfrac{\partial P}{\partial y}=2x$ and $\dfrac{\partial Q}{\partial x}=2x$. Since $P,$ $Q,$ $\dfrac{\partial P}{\partial y},$ and $\dfrac{\partial Q }{\partial x}$ are continuous everywhere in the $xy$-plane and since $\dfrac{ \partial P}{\partial y}=\dfrac{\partial Q}{\partial x}$, then $\mathbf{F}=\mathbf{F}(x,y)=(2xy+24x)\mathbf{i}+(x^{2}+16)\mathbf{j}$ is a conservative vector field. By the Fundamental Theorem of Line Integrals, $\int_{C}\mathbf{F}\,{\cdot}\, d\mathbf{r}$ is independent of the path.

(b)Since $\mathbf{F}=\mathbf{F}(x,y)=(2xy+24x)\mathbf{i}+(x^{2}+16)\mathbf{j}$ is a conservative vector field, there is a potential function $f$ for $\mathbf{F}$ for which $$\nabla\! \ f=\mathbf{F}=(2xy+24x)\mathbf{i}+(x^{2}+16)\mathbf{j}$$

Since $\nabla\! \ f=\dfrac{\partial f}{\partial x}\mathbf{i}+\dfrac{\partial f}{\partial y}\mathbf{j}$, then $$\begin{equation*} \dfrac{\partial f}{\partial x}=2xy+24x\qquad \hbox{and}\qquad \dfrac{\partial f}{\partial y}=x^{2}+16 \end{equation*}$$

We integrate the first of these partial derivatives partially with respect to $x$. $$\begin{equation*} f(x,y)=\int (2xy+24x)\,dx=x^{2}y+12x^{2}+h(y) \end{equation*}$$

where the “constant of integration,” $h(y)$, is a function of $y$. Now we differentiate $f$ with respect to $y$, obtaining $$\begin{equation*} \dfrac{\partial f}{\partial y}=\dfrac{\partial }{\partial y}[x^{2}y+12x^{2}+h(y)] =x^{2}+h' (y) \end{equation*}$$

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But $\dfrac{\partial f}{\partial y}=x^{2}+16$, so $h' (y)=16$. Now we integrate $h'$ with respect to $y$; then $$h(y)=16y+K$$

where $K$ is a constant. So, $$f(x,y)=x^{2}y+12x^{2}+16y+K$$

is a potential function for $\mathbf{F}.$

(c) Since $\mathbf{F}$ is independent of the path, we can use the Fundamental Theorem of Line Integrals. Since $f(x,y)=x^{2}y+12x^{2}+16y + K$ and $\mathbf{F}=\nabla\! \ f,$ $$\begin{equation*} \int_{C}\mathbf{F}\,{\cdot}\, d\mathbf{r}=\int_{C}\nabla\! \ f\,{\cdot}\, d\mathbf{r}=\big[ f(x,y)\big] _{(0,1)}^{(1,2)}=f(1,2)-f(0,1)=46-16=30 \end{equation*}$$

for any piecewise-smooth curve $C$ connecting $( 0,1)$ to $( 1,2)$.