In the figure below let be the area of the shaded region and
the area of the triangle
. Compute
-
;
-
;
-
.
- First, to calculate the area of the triangle
we calculate the area of the polygon
and subtract the area of the triangle
. Since the polygon
is two congruent triangles (since the line from
to
bisects the polygon into two equal sized triangles) we have
To calculate the area of the triangle
we use the auxiliary point
. The line from
to
is the height of the triangle
and it has length
. Therefore,
Hence,
- The area of the shaded region
is the area of circular sector minus the area of the triangle
(which we calculated in part (a) as
). The area of the sector is given by
Hence,
- For this we take the limit using our expressions for
and
obtained in parts (a) and (b) and use L’Hopital’s,
Then, we note that
(from the trig identity
). Therefore,
We have the following diagram, with some additional markings from the diagram in Apostol,