[tex]\int_{1+0i}^{0+1i}\: zdz\: =\int_1^i\: zdz\: =[\frac{z^2}{2}]_1^i\: =\frac{\: i^2}{2}\: -\frac{\: 1^2}{2}\: =\: \frac{-1}{2}\: -\frac{\: 1}{2}\: =\: -1[/tex]

[tex]z=e^{i\theta}[/tex]

[tex]dz=i\: e^{i\theta}\: d\theta[/tex]

[tex]\int_c\: zdz\: =\: \int_0^{\frac{\pi}{2}}\: (e^{i\theta})(ie^{i\theta}d\theta)\: =\: \: i\: \int_0^{\frac{\pi}{2}}\: e^{2i\theta}\: d\theta\: \: =\: \frac{\: i}{2i}[e^{2i\theta}]_0^{\frac{\pi}{2}}\: =\frac{\: 1}{2}\: (e^{i\pi}\: -\: 1)\: =\frac{\: 1}{2}\: (\cos\pi\: +\: i\: \sin\pi\: -1)\: =\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \frac{\: 1}{2}\: (-1+0-1)\: =\: -1[/tex]