The question is to find the shaded area between the rectangle and the top part of the circle.
Let's see if Geogebra can help.
1)Draw 4 points (0,0),(0,.53)(1.53,.53)(1.53,0) and call them A,B,C and D.
2)Draw point E at (.765, .35)
3)Use the circle through 3 points tool and draw a circle through A,E and D
4)Draw a line through E and perpendicular to the x axis.
5)Find the point of intersection between the perpendicular line and the circle. Call the bottom point H.
6)Find the midpoint between E and H and call it I. This is the midpoint of the circle.
7)Measure IH with the measuring tool. That is the radius of the circle. You should get 1.01104
8)Measure the angle DIA with the angle tool, you should get 98.33948 degrees.
Now we can use the area above a chord formula
Plugging in r = 1.01104 and θ = 98.33948 we get 0.37153. Now we subtract from .8109 - 0.37153 = 0.439369, which is close. See the drawing to check yours.