If $y = \frac{b}{a}\sqrt{a^2-x^2}$ is the top half of the ellipse, shouldn't the function for the ellipse in the first quadrant be half the area of the top half of the ellipse? Why is it $1/4$ of the area if the function given is for the top half of the ellipse?
The area enclosed by the ellipse consists of four equal parts, one in each quadrant. Therefore the area in the first quadrant is one quarter of the total area.