$$
\int_{\Upsilon}e^z dz
$$
$$
\text{line from 0 to } z_{0}$$
Therefore
$$
r(t)= tz_{0}
$$
$$
r'(t) = z_{0} (0\leq t \leq 1)
$$
And since
$$
f(z)=e^z
$$
$$
f(r(t))= e^{z_{0}}
$$
So
$$
\int_{\Upsilon}e^z dz = \int_{0}^{1}f(r(t))r'(t)dt = \int_{0}^{1}e^{tz_{0}}z_{0} dt = z_{0}(\frac{1}{z_{0}}e^{z_{0}}- \frac{1}{z_{0}})= e^{z_{0}} - 1
$$
