I'm not sure what you're doing but you want the line integral

of(x^2)*y*sqrt(z)dz on the curve x=t^3, y=t, z=t^2 ?

F= (x^2)*y*sqrt(z)k

Since

x=t^3, y=t, z=t^2

F= t^6*t*tk= t^8k

r =t^3i+ tj+ t^2k

r ' =3t^2i+j+ 2tk

F*r ' =2t^9

Which is easy to integrate over the interval 0 to 1