I would have done the same as what you have; could the lower bound at the very last step have been changed in the book from -pi/2 to 0, which would then require to double the integral in order to get the full loop?
I am doing a question on finding the area enclosed by a loop, given by the equation:
with .
I know that the area is
Which works out to be
Taking out the 4 and expanding I end up with
The book however has a 4 outside the brackets, is this a typo or something that I've missed?
Thanks for the help