Green's Theorem
We need to find the Area inside x32+y32=1
Solution:
Given,
x32+y32=1
Let
x=cos3 t and y=sin3 t, 0≤t≤2π
By Green's theorem
Area inside =
21∮A(x dy−y dx)
We know,
x=cos3 t
dx=3 cos2 (t)×(−sin(t))=−3cos2(t)sin(t) dt
y=sin3 t
dy=3sin2(t)×cos(t) dt
Now plug all these in the formula of Green's theorem
Area inside = 21∮A((x dy−y dx)=21[∫t=02π(cos3 t3sin2(t)×cos(t) dt)−sin3 t(−3cos2(t)sin(t)) dt)]
=23[∫t=02π(cos4 t sin2(t)× dt)+sin4 t cos2(t) dt)]
=23[∫t=02πcos2 t sin2(t)× (cos2t+sin2 t)]dt
=23∫t=02πcos2 t sin2(t)dt
=23×2×2∫t=02πcos2 t sin2(t)dt
=6∫t=02πcos2 t sin2(t)dt
=46∫t=02π(2sintcost)2dt==46∫t=02πsin2 2t dt
( Since sin 2t = 2 sint t cos t)
=46∫t=02π21−cos4t dt=43[t−4sin4t]t=0t=2π
=43×2π=83π
Answer: The Area inside = 83π
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