计算∫∫(区域D)xy눀dσ,其中D是由圆周x눀+y눀=4及y轴所围成的右半闭区域。

2025-01-21 14:16:10
推荐回答(1个)
回答1:

右边半圆:x = √(4 - y²)
∫∫_D (xy²) dσ
= ∫(-2~2) dy ∫(0~√(4 - y²)) xy² dx
= ∫(-2~2) x²y²/2 |(0~√(4 - y²)) dy
= 2∫(0~2) (4 - y²)y²/2 dy
= ∫(0~2) (4y² - y⁴) dy
= (4/3)y³ - (1/5)y⁵ |(0~2)
= (4/3)(8) - (1/5)(32)
= 64/15