What is the Correct Way to Find the Length of a Contour Gamma?

[stunner5000pt]

Find the length of the contour Gamma parametrized by $z = z(t) = 5e^{3it}$

the legnth of contour is $$\int_{0}^{\pi} \left|\frac{dz}{dt}\right| dt$$

now $$\left|\frac{dz}{dt}\right| = |15ie^{3it}| = |15ie^{3t}\cos(t) - 15e^{3t}\sin(t)| = \sqrt{225e^{6t} + 225e^{6t}}$$

is this right so far?? I think I am making a mess of the magnitude part

 

[nrqed]

Find the length of the contour Gamma parametrized by $z = z(t) = 5e^{3it}$

the legnth of contour is $$\int_{0}^{\pi} \left|\frac{dz}{dt}\right| dt$$

now $$\left|\frac{dz}{dt}\right| = |15ie^{3it}| = |15ie^{3t}\cos(t) - 15e^{3t}\sin(t)| = \sqrt{225e^{6t} + 225e^{6t}}$$

is this right so far?? I think I am making a mess of the magnitude part

The magnitude of $15ie^{3it}$ is 15! (just take the expression times its complex conjugate and then take the square root!).

Your second step is all wrong. You seem to think that $e^{it} = e^t e^i$ ?? That`s of course incorrect!

Euler`s identity applied to $e^{3 i t}$gives $cos(3t) + i sin(3t)$

EDIT: which can also be written as $(e^{it})^3 = (cos t + i sin t)^3$ of course

 

[stunner5000pt]

The magnitude of $15ie^{3it}$ is 15! (just take the expression times its complex conjugate and then take the square root!).

Your second step is all wrong. You seem to think that $e^{it} = e^t e^i$ ?? That`s of course incorrect!

Euler`s identity applied to $e^{3 i t}$gives $cos(3t) + i sin(3t)$

EDIT: which can also be written as $(e^{it})^3 = (cos t + i sin t)^3$ of course

dont i feel stupid!
i was using Euler's identitiy incorrectly, when my course... for the most part... is based on it!