[-a,a] ∫ (x²-x)√(a²-x²) dx
=[-a,a] ∫ x²√(a²-x²) dx - [-a,a] ∫ x√(a²-x²) dx （后半部分是奇函数,在对称区间的定积分为零）
=[0,a] 2 ∫ x²√(a²-x²) dx
=[0,a] 2 ∫ (a²-a²+x²)√(a²-x²) dx
=[0,a] 2 ∫ a²√(a²-x²) - (a²-x²)^(3/2) dx
= a²x√(a²-x²)+ a^4 arctan[x/√(a²-x²)] - ¼ x(5a²-2x²)√(a²-x²) - ¾ a^4 arctan[x/√(a²-x²)] | [0,a]
=¼ (2x²-a²)x√(a²-x²)+¼ a^4 arctan[x/√(a²-x²)] | [0,a]
=(x→a)lim¼ a^4 arctan[x/√(a²-x²)]
=(u→π/2)lim¼ u a^4 （令 x=asinu,则arctan[x/√(a²-x²)]=u）
=(πa^4)/8