D由y=x,y=x÷2,y=2组成,求x^2+y^2-y的二重积分,结果为32/3,

Question

D 由y=x,y=x÷2,y=2组成,求x^2+y^2-y的二重积分,结果为32/3,

Answer 1

宜用Y型区域：
∫∫ (x² + y² - y) dσ
= ∫(0→2) dy∫(y→2y) (x² + y² - y) dx
= ∫(0→2) [x³/3 + xy² - xy] |(y→2y) dy
= (1/3)∫(0→2) (10y³ - 3y²) dy
= (1/3)[10y⁴/4 - y³] |(0→2)
= (1/3)(32)
= 32/3