How to parametrize this contour integral

Question

I am trying to calculate the following integral: $$\int_Cz^2dz$$ where $C$ is the line from $1$ to $i$. If these were real numbers, I would use the equation for that line, $y=-x+1$. However, this doesn't work in this case. I tried parametrizing as follows: $$z=x+iy=(t-1)+ti$$ This matches the first point, but not the second, as $t=i$ is $z=i-1-1=i-2$ rather than just i, as I wanted. How can I correctly parametrize this, and know how to parametrize these situations in the future?

Answer 1

A line from point $A$ to point $B$ can be parametrized as $(1-t)\cdot A + t\cdot B$. This expression clearly equals to $A$ when $t=0$ and to $B$ when $t=1$.

In particular, therefore, $(t-1)+ti$ is what you want, and since you are only selecting $t\in[0,1]$, you don't really need to bother about what happens when $t=i$.