(x^2 +y^2)^3 = (a^2)*(x^2)*(y^2)
[m]x=rcosθ[/m]
[m]y=rsinθ [/m]
[m]x^2+y^2=r^2[/m]
[m](r^2)^3=a^2\cdot (rcosθ)^2\cdot (rsinθ )^2[/m] ⇒ [m]r^6=r^4\cdot a^2\cdot (cosθ)^2\cdot (sinθ )^2[/m]
[m]r^2= a^2\cdot (cosθ)^2\cdot (sinθ )^2[/m]
[m]r= a\cdot cosθ\cdot sinθ [/m]
[m]r= \frac{a}{2}\cdot sin2θ [/m]
[m] sin2θ ≥ 0 [/m] ⇒ [m]0 +2πk ≤ 2 θ ≤ π+2πk, k ∈[/m] ⇒ [m]0+πk ≤ θ ≤ \frac{π}{2}+πk, k ∈ [/m][b]Z[/b]
Кривая в 1 и 3 четверти
[m]\frac{1}{2}S=\frac{1}{2} ∫^{\frac{π}{2}} _{0}( \frac{a}{2}\cdot sin2θ)^2d θ= [/m]