What is the arclength of $r=4theta$ on $theta in [-pi/4,pi]$ ?

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Answer 1 · 2016-09-05T12:43:54

$approx 27.879$

This is an outline method. The grind of some of the work has been done by computer.

Arc length $s = int dot s \ dt$

and $dot s = sqrt (vec v * vec v)$

Now, for $vec r = 4 theta \ hat r$

$vec v = dot r hat r + r dot theta hat theta$

$= 4 dot theta \ hat r + 4 theta dot theta \ hat theta$

$= 4 dot theta ( hat r + theta \ hat theta )$

So $dot s = 4 dot theta sqrt(1 + theta ^2)$

Arc length $s = 4 int_(t_1)^(t_2) sqrt(1 + theta ^2) \ dot theta \ dt$

$= 4 int_(-pi/4)^(pi) sqrt(1 + theta ^2) \ d theta$

$= 2 [ theta sqrt(theta^2+1) +sinh^(-1) theta ]_(-pi/4)^(pi)$ computer solution. See Youtube linked here for the method

$approx 27.879$ computer solution

What is the arclength of r=4theta r=4theta on theta in [-pi/4,pi]theta in [-pi/4,pi]?