The area of the interior is mathematically given as
[tex]3 \pi/8[/tex]
This is further explained below.
Generally, the equation for the curve is mathematically given as
x2/3 + y2/3=1
parametrizing of the curve, Where
(cos3t)2/3+(sin3t)2/3=1 implies (cos2t)+(sin2t)=1 ,for 0<=t<=2pi
dx=-3cos2t*(sint) dt,dy=3sin2t cost dt
so the area equals (By Green's theorem)
1/2* integration of (-ydx+xdy) over C
[tex]=1/2*integration of(0 to 2pi) (-sin3t(-3cos2t*(sint))+cos3t(3sin2t cost)) dt\\\\=1/2*integration of(0 to 2pi) (3sin4t cos2t+3cos4tsin2t) dt\\\\=3/8*integration of(0 to 2pi) [sin2(2t)]dt[/tex]
Hence
plug sin2(2t)=(1-cos(4t))/2
[tex]=3/8*integration\ of\ (0\ to\ 2 \pi) [(1-cos(4t))/2]dt\\\\=3/16*[2 \pi] since sin(8pi)=0\\\\=(3 \pi)/8[/tex]
In conclusion, the area of the interior of the curve is
[tex]3 \pi/8[/tex]
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