Parameterize the path [tex]\mathcal C[/tex] by
[tex]\mathbf r(t)=x(t)\,\mathbf i+y(t)\,\mathbf j[/tex]
with
[tex]x(t)=3\cos t[/tex]
[tex]y(t)=3\sin t[/tex]
and [tex]0\le t\le\pi[/tex]. Then
[tex]\displaystyle\int_{\mathcal C}\mathbf f(x,y)\cdot\mathrm d\mathbf r=\int_{t=0}^{t=\pi}\mathbf f(x(t),y(t))\cdot\dfrac{\mathrm d\mathbf r}{\mathrm dt}\,\mathrm dt[/tex]
[tex]=\displaystyle\int_0^\pi(9\cos^2t\,\mathbf i+9\sin^2t\,\mathbf j)\cdot(-3\sin t\,\mathbf i+3\cos t\,\mathbf j)\,\mathrm dt[/tex]
[tex]=27\displaystyle\int_0^\pi(\cos t\sin^2t-\sin t\cos^2t)\,\mathrm dt=-18[/tex]
