Evaluate $\int _C f$ where $f(z)=x^2+iy^2$ and where $C$ is given by $z(t)=t^2+it^2, 0\leq t \leq 1$.
I tried reading an example in the book, using the formula $\int_C f(z)dt=\int_a^bf(z(t))\dot{z}(t)dt$, but I am not sure how to proceed. I know that $\dot{z}(t)=2t+2it$ but I do not understand what is mean by $f(z(t))$. This is the first time I have seen composition of complex functions. Is it as simple as substituting the real part of $z(t)$ into the real part of $f$ and similarly with the imaginary parts? That is, does $f(z(t))=t^4+it^4$? Thank you for your help.