I'm trying to solve this problem with calculus.
A string is wound symmetrically around a circular rod. The string goes exactly 4 times around the rod. The circumference of the rod is 4 cm. and its length is 12 cm. Find the length of the string.
My parametrization is
$r(t)=\left(t, \frac{2}{\pi}\cos(t),\frac{2}{\pi}\sin(t)\right)$.
When I compute the integral
$\int_0^{12} \sqrt{1+(-\frac{2}{\pi}\sin(t))^2+(\frac{2}{\pi}\cos(t))^2}$,
my answer does not match the correct answer (20 cm). I suspect the problem is with the period of the trig functions, but I do not know what has to be done with this. Thanks.
Your $r(t)$ moves by $2\pi$ in the $x$ direction per turn, whereas it should move by $3$, i.e. the $x$ component should be $3t/(2\pi)$, and the upper integration limit should then be $8\pi$. Alternatively, you could make the argument of the trigonometric functions $2\pi t/3$ (without changing the upper integration limit).
That makes the result come out right, but there's a much easier way to do this: A cylinder is a developable surface, so you can just unroll it as a plane and calculate the length of the string by Pythagoras as
$$ \sqrt{(12\,\text{cm})^2+(4\cdot4\,\text{cm})^2}=20\,\text{cm}\;. $$