I was asked to evaluate the following contour integral: $$\int_C\operatorname{Im}(z)\,dz$$ where $C$ is the straight line segment from $z = i$ to $z = 1+2i$.
My solution for this problem is the following:
First define $z = x+iy$. Then we have $\operatorname{Im}(z) = y$ and we have the integral $\int_Cy(dx+i\,dy)$. So we have $$\int_Cy\,dx +i\int_Cy\,dy = \int_0^1(x+1)\,dx + i\int_1^2y\,dy = \frac{3}{2}+\frac{3}{2}i$$ because the line segment from $z = i$ to $z = 1+2i$ is part of the line $y = x+1$ so I simply put $x+1$ in place of $y$. What I am unsure about is that is it mathematically correct to put $x+1$ in place of $y$ directly or should I make a parametrization like using another variable $t$ and then convert the integral to another integral with respect to $t$. I hope my solution is correct as well. Thank you in advance.
What you've done is correct, and if you use a parameter $t,$ of which $x$ and $y$ are both functions, then there's no reason why you wouldn't have $x=t$ and $y=t+1,$ and since $x$ is the same thing as $t,$ you can just call it $x$ rather than $t$ and then you have exactly what you wrote.
(I wonder if that's the longest sentence I've written in a posted answer? Probably not.)