$$\sqrt{(x^{\prime }{)}^2+(y^{\prime }{)}^2}dt=a\sqrt{(1-\cos t{)}^2+{\sin }^{2}t}dt=2a\sin \frac{t}{2}dt$$

то

$$\begin{gathered} \underset{C}{\int }{y}^{2}dl=\underset{a}{\overset{b}{\int }}f(x(t),y(t))\sqrt{(x^{\prime }{)}^2+(y^{\prime }{)}^2}dt= \\ =2a^3\underset{0}{\overset{2\pi }{\int }}(1-\cos 2t{)}^2\sin \frac{t}{2}dt= \\ =-16a^3\underset{0}{\overset{2\pi }{\int }}\left(1-{\cos }^2\frac{t}{2}+{\cos }^4\frac{t}{2}\right)d\left(\cos \frac{t}{2}\right)=\frac{256}{15}{a}^3 \end{gathered}$$