There are two ways to look at this. If you were to "cut" the hoop and straighten it out, it would be a rectangle of width dx and length equal to thecircumferenceof the circle, . The area is, then, .

Or: think of the hoop as the areabetweentwo circles, one of radius r and the other of radius r+ dx. The larger circle has area and the smaller area . The difference is but since dx is a "differential", we ignore its square leaving just .