Compute line integral: $\int_{r} Re(z) \,dz$, where r is line segmemt from 1 to i
$$
f(z) = Re(z)
$$
$$
r(t) = (1-t)z_0 + tz_1 = (1-t)*1 + ti = 1-t+it , 0<= t <= 1
$$
$$
r^{'}(t) = i-1
$$
$$
f(r(t)) = Re(r(t)) = 1-t
$$
$$
\int_{r} f(z) \,dz = \int_{0}^{1}f(r(t))r^{'}(t) \,dt
$$
$$
=\int_{0}^{1}(1-t)(i-1) \,dt
$$
$$
=(i-1)*(t-\tfrac{1}{2} t^2)\Big|_0^1
$$
$$
=(i-1)(1-\tfrac{1}{2}-0)
$$
$$
=\tfrac{1}{2}(i-1)
$$
