I tried following the example in my book, and I got stuck when trying to take the integral for the second part with $y = x^2$. I'm not completely confident on what to parameterize with here. I tried replacing $x^2$ with $y$, then in the radical I had $4 + y^3$, but when trying to integrate that with u substitution I couldn't finish it. I'm not sure I'm setting everything up right.
I'm also not sure how to tell which integral goes with which segment. Do you just assume the first goes with the first and second with the second?
hint
the first segment can be parametrised as
$$x=0+2t \;\;\;,\;\; y=0+t $$
the integral along this segment is
$$\int_0^1 \Bigl ((2t+2t)(2dt)+4t^2 (dt)\Bigr)=$$
$$4\int_0^1t (2+t)dt=\frac {16}{3}$$
You can do it for the second segment defined by
$$x=2+t \;\;\;, \;\; y=1-t$$