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

1 Answer
Jan 10, 2018

Answer:

#(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.