Show that the area of the parabolic sector equals to that of the parabolic rectangle

Question

question

The blue line is the directix and the blue point is the focal point of the parabola.

Reviewer's Note:

Taking points at vertex and latus rectum of unit focal length

Red area $= \frac43 $ and the Blue area = $2\cdot 2 - \frac43 = \frac83$, the assetion appears to me incorrect. Please check these. After checking please comment and I shall delete my edit. Hope it is in order.

Answer 1

Drop a perpendicular line from the focal point to the directrix. Find the area of parabolic rectangle (green area) in two ways:

1) Half of the difference between areas of large and small trapeziums.

2) the area under the parabola minus the area under directrix.

The two areas must be equal, however my calculation shows they are not.

You can check with $y=\frac14x^2$ with vertex $(0;0)$ and focal length $a=1$. Consider the interval $(1,2)$. 1) $\frac{19}{16}$. 2) $\frac{19}{12}$.