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 areaoutsideand inside .