Integral along parametric curve

Question

Studying complex analysis I came across the following example. Consider the function $f(z)=x^2+i\,y^2$ integrated along the smoof curve parametrized by $z(t)=t+i\,t$ for $0\leq t\leq1$. How do I construct this parametric curve?

I know that such a curve $z(t)$, must be in the complex plane, and that, I can start that the start and end points are, respectively, $z(0)=0$ and $z(1)=1+i$. But I really can not build the curve itself. I just need to understand this to continue with the calculation of the integral of $f(z)$ on the aforementioned curve, which by the way is quite simple in this case. In advance I am grateful for the clarifications.

Answer 1

You can think at the complex numbers as points in the Argand's plane.

So the line $z(t)=t+it$ is the line represented parametrically by $(x,y)=(t,t)$ with $ 0\le t \le 1$, that is the segment from the origin to the point $(1,1)$.

Answer 2

Also you can think the following idea. Writing the curve one has $$z(t) = t+it=t(1+i), $$i. e., it's a segment that starts at the origin with direction vector $$1+i. $$