The elemental area, denoted by the shaded region,
is
found by taking the area to the the
3sin
r
θ
=
curve and
subtract
the area to the
2
sin
r
θ
=
−
curve.
The area integral is thus:
/2
2
2
/6
1
2
(3sin
)
(2
sin
)
2
A
d
π
π
θ
θ
θ
=
−
−
∫
Note that the factor “
2
” is used because we need to double
the result to find the total area by virtue of symmetry.
After some algebra, the integral becomes:
/2
/2
/2
2
/6
/6
/6
8
sin
4
sin
4
A
d
d
d
π
π
π
π
π
π
θ
θ
θ
θ
θ
=
+
−
∫
∫
∫
You can complete the problem from here.

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