Question

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

Answer 1

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

Explanation:

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.