Question

计算二重积分∫ ∫ xy^2dxdy,D是半圆区域:x^2+y^2≤4,x≥0

Answer 1

第一题：

x² + y² = 4,x ≥ 0 ==> x = √(4 - y²)

∫∫_D xy² dxdy

= ∫(-2-->2) dy ∫(0-->√(4 - y²)) xy² dx

= 1/2 · ∫(-2-->2) dy x²y² |(0-->√(4 - y²))

= ∫(0-->2) (4 - y²)y² dy

= ∫(0-->2) (4y² - y⁴) dy

= (4/3)y³ - (1/5)y⁵ |(0-->2)

= (4/3)2³ - (1/5)2⁵

= 64/15

第二题：

y = x²,y = 2,交点(- √2,2),(√2,2)

面积A = 2∫(0-->2) √y dy = 2 · (2/3)y^(3/2) |(0-->2)

= (4/3)(2)^(3/2)

= (8√2)/3

旋转体积V = π · ∫(0-->2) x² dy

= π · ∫(0-->2) y dy

= π/2 · y² |(0-->2)

= π/2 · 4

= 2π