It's the red segment.
Plug and play
#A = 1/2 \int_{\theta = pi/8}^{pi/4} \ r^2(theta) \ d theta #
# = \int_{\theta = pi/8}^{pi/4} \ 2sin^2 (4theta+(11pi)/12) \ d theta#
using # cos 2A = 1 - 2 sin^2 A#
# = \int_{\theta = pi/8}^{pi/4} \ 1 - cos (8theta+(11pi)/6) \ d theta#
# = [ \ theta - 1/8 sin (8theta+(11pi)/6) ]_{\theta = pi/8}^{pi/4}#
# = [ \ pi/4 - 1/8 sin (2 pi+(11pi)/6) ] - [ \ pi/8 - 1/8 sin ( pi+(11pi)/6) ]#
# = pi/8 + 1/8 [ sin ( pi+(11pi)/6)- sin (2 pi+(11pi)/6) ]#
# = pi/8 + 1/8 [ sin ( pi) cos ((11pi)/6) + cos ( pi) sin ((11pi)/6) - sin (2 pi) cos ((11pi)/6) - cos (2 pi) sin ((11pi)/6) ]#
# = pi/8 + 1/8 [ (-1) (- 1/2) - (1)(-1/2) ]#
# = (pi + 1)/8#
