What is the arc length of the polar curve $f(theta) = 2costheta-theta$ over $theta in [pi/8, pi/3]$ ?

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Answer 1 · 2018-01-10T16:36:18

$(21pi)/8 -sqrt3 /2 -2 +1/sqrt2 +4 cos pi/8$

The given function is $r= 2cos theta - theta$

$(dr)/(d theta)$ would be $-2sin theta -1$

Arc length formula is $int_a^b 1/2 r^2 d theta$

The required arc length would be $1/2 int_(pi/8) ^(pi/3) (4sin^2 theta + 4 sin theta +1) d theta$

= $1/2 int_(pi/8)^(pi/3) (2-2cos 2theta + 4sin theta +1)$

= $[3 theta - sin 2 theta- 4 cos theta]_(pi/8)^(pi/3)]$

= $3pi -sin( (2pi)/3)-4 cos (pi/3) -(3pi)/8 + sin (pi/4) +4 cos (pi/8)$

= $(21pi)/8 -sqrt3 /2 -2 +1/sqrt2 +4 cos pi/8$

Further calculations may be made as desired.

What is the arc length of the polar curve f(theta) = 2costheta-theta f(theta) = 2costheta-theta over theta in [pi/8, pi/3] theta in [pi/8, pi/3] ?