Parameterize [tex]C[/tex] by
[tex]\mathbf r(t)=\langle x(t),y(t)\rangle=\langle3\cos t,3\sin t\rangle[/tex]
where [tex]0\le t\le\pi[/tex]. Then the line integral is
[tex]\displaystyle\int_Cx^2y\,\mathrm dS=\int_{t=0}^{t=\pi}x(t)^2y(t)\left\|\frac{\mathrm d\mathbf r(t)}{\mathrm dt}\right\|\,\mathrm dt[/tex]
[tex]=\displaystyle\int_{t=0}^{t=\pi}(3\cos t)^2(3\sin t)\sqrt{(-3\sin t)^2+(3\cos t)^2}\,\mathrm dt[/tex]
[tex]=\displaystyle3^4\int_{t=0}^{t=\pi}\cos^2t\sin t\,\mathrm dt[/tex]
Take [tex]u=\cos t[/tex], then
[tex]=\displaystyle-3^4\int_{u=1}^{u=-1}u^2\,\mathrm du[/tex]
[tex]=\displaystyle3^4\int_{u=-1}^{u=1}u^2\,\mathrm du[/tex]
[tex]=\displaystyle2\times3^4\int_{u=0}^{u=1}u^2\,\mathrm du[/tex]
[tex]=54[/tex]
