$\int_\Gamma\Re(z)\,dz$ around the unit square

Write \(\Gamma = \sum_{1\leq k \leq 4}\gamma_k\), starting at zero and traversing counterclockwise through \(0\), \(1\), \(1+i\), \(i\):

For \(t\in[0,1]\):

• \(\gamma_1(t)=t\): \(\int_0^1 t\dt = 1/2\).

• \(\gamma_2(t)=1+it\): \(\int_0^1 1\cdot i \dt = i\).

• \(\gamma_3(t)=(1-t)+i\): \(-\int_0^1 (1-t)\dt = -1/2\).

• \(\gamma_4(t)=i(1-t)\): \(-i\int_0^1 0 \dt = 0\).

So \(\int_\Gamma \Re(z) \dz = i\).