[tex]\dfrac{\partial f}{\partial x}=7+8xy^2[/tex]
[tex]\dfrac{\partial f}{\partial y}=8x^2y[/tex]
The first equation gives
[tex]f(x,y)=7x+4x^2y^2+g(y)[/tex]
Differentiating with respect to [tex]y[/tex] gives
[tex]8x^2=8x^2y+\dfrac{\mathrm dg}{\mathrm dy}\implies\dfrac{\mathrm dg}{\mathrm dy}=0\implies g(y)=K[/tex]
for some constant [tex]K[/tex]. So
[tex]f(x,y)=7x+4x^2y^2+K[/tex]
and by the fundamental theorem of calculus,
[tex]\displaystyle\int_C\nabla f\cdot\mathrm d\vec r=f\left(3,\frac13\right)-f(1,1)=25-11=\boxed{14}[/tex]
