Let [tex]x=\cos t[/tex] and [tex]y=\sin t[/tex]. Then [tex]C[/tex] can be parameterized by
[tex]\vec r(t)=\cos t\,\vec\imath+\sin t\,\vec\jmath+(\sin^2t-\cos^2t)\,\vec k[/tex]
with [tex]0\le t\le2\pi[/tex], and its derivative is
[tex]\dfrac{\mathrm d\vec r}{\mathrm dt}=-\sin t\,\vec\imath+\cos t\,\vec\jmath+4\sin t\cos t\,\vec k[/tex]
Now,
[tex]\vec F(x,y,z)=x^2y\,\vec\imath+\dfrac{x^3}3\,\vec\jmath+xy\,\vec k[/tex]
[tex]\implies\vec F(\vec r(t))=\cos^2t\sin t\,\vec\imath+\dfrac{\cos^3t}3\,\vec\jmath+\cos t\sin t\,\vec k[/tex]
Then the work done by [tex]\vec F[/tex] along [tex]C[/tex] is
[tex]\displaystyle\int_C\vec F(x,y,z)\cdot\mathrm d\vec r=\int_0^{2\pi}\vec F(\vec r(t))\cdot\frac{\mathrm d\vec r}{\mathrm dt}\,\mathrm dt=\int_0^{2\pi}\left(3\cos^2t\sin^2t+\frac{\cos^4t}3\right)\,\mathrm dt=\boxed{\pi}[/tex]
