令X=R*sina
那么dx=R*cosa da
而a的范围为(-π/2,π/2)
所以
原积分
=∫(-π/2到π/2) √(R²-R²sin²a) *Rcosa da
=∫(-π/2到π/2) cosa *R²cosa da
=R² * ∫(-π/2到π/2) cos²a da
=0.5R² * ∫(-π/2到π/2) cos2a+1 da
=0.5R² *(0.5sin2a+a) 代入上下限π/2和 -π/2
=0.5R² *[0.5sinπ+π/2- 0.5sin(-π) -(-π/2)]
=0.5πR²
于是解得
原积分=0.5πR²