Proposition

Compute the lengths of the paths from this post:

The circle $C[1 + i, 1]$, oriented counter-clockwise.
The line segment from $-1 - i$ from $2i$.
The top half of the circle $C[0, 34]$, oriented clockwise.
The rectangle with vertices $\pm 1 \pm 2i$, oriented counter-clockwise.

Solution

1

$\gamma(t) = 1 + i + e^{2\pi it}$ with $0 \leq t \leq 1$.

$\int_{0}^{1} \abs{\gamma’(t)} dt = 2\pi \int_0^1 1 dt = 2\pi$.

2

$\gamma(t) = t + (3t + 2)i$ with $-1 \leq t \leq 0$. $\int_{-1}^{0} \abs{\gamma’(t)} dt = \int_{-1}^{0} \sqrt{1 + 9}dt = \sqrt{10}$.

3

$\gamma(t) = 34e^{-2\pi it}$ with $1/2 \leq t \leq 1$. $\int_{1/2}^{1} \abs{34e^{-2\pi it} \cdot -2\pi i} dt = 68\pi / 2 = 34\pi$.

4

\[\begin{align*} \gamma(t) &= \begin{cases} 1 + (t - 2)i & (0 \leq t \leq 4) \\ 1 - (t - 4) + 2i & (4 \leq t \leq 6) \\ -1 + (2 - (t - 6))i & (6 \leq t \leq 10) \\ -1 + (t - 10) - 2i & (10 \leq t \leq 12). \end{cases} \end{align*}\]

$\int_{0}^{4} \abs{i} dt + \int_{4}^{6} \abs{-1} dt + \int_{6}^{10} \abs{-i} dt + \int_{10}^{12} \abs{1} dt = 12$