The area swept out consists of two circular sectors of different radii. The right and left radii are, respectively,
(1-l) l /2  and   (1+l) l /2.
Noting that one of the signed areas is positive and the other is negative, we get
A = 1/2 [(1-l) l /2] 2 Dq  -  1/2 [(1+l) l /2] 2 Dq.
Multiplying this out and simplifying yields A = -1/2 l l 2 Dq.