Believe it or not, your book is right. The area swept out by a radius r through angle is given by so your differential of area is
For from 0 to , so the radius only extends from 0 to . For from to is less than so the radius extends only to that.
The area between the two curves in the first quadrant is .
Of course, there is a symmetrical part in the second quadrant so that is multiplied by 2.
What you appear to be calculating is the area outside and inside .