Question

计算二重积分∫∫D根号（4-x²-y²）dxdy,其中D为以X的平方+Y的平方=2X为边界的上半圆域

Answer 1

x = rcosθ，y = rsinθ

x² + y² = 2x
(rcosθ)² + (rsinθ)² = 2rcosθ
r²(cos²θ + sin²θ) = 2rcosθ
r = 2cosθ
∫∫_D √(4 - x² - y²) dxdy
= ∫(0，π/2) ∫(0，2cosθ) √(4 - r²) * r drdθ
= (- 1/3)∫(0，π/2) (4 - r²)^(3/2) |(0，2cosθ) dθ
= (- 1/3)∫(0，π/2) [(4 - 4cos²θ)^(3/2) - (4 - 0)^(3/2)] dθ
= (- 8/3)∫(0，π/2) |sinθ|³ dθ + (8/3)∫(0，π/2) dθ
= (- 8/3)∫(0，π/2) sin³θ dθ + (8/3)(π/2 - 0)
= (- 8/3)∫(0，π/2) sin²θ d(- cosθ) + 4π/3
= (8/3)∫(0，π/2) (1 - cos²θ) d(cosθ) + 4π/3
= (8/3)[cosθ - (1/3)cos³θ] |(0，π/2) + 4π/3
= (8/3)(0 - 2/3) + 4π/3
= (4/9)(3π - 4) ≈ 2.41101

Answer 2

极坐标变换：x＝rcosa，y＝rsina，x^2+y^2<=2x等价于
r^2<=2rcosa，故0<=r<=2cosa，
上半圆域对应着0<=a<=pi/2。
Jacobian行列式为r，4－x^2－y^2＝4－r^2，于是原积分
＝积分(从0到pi/2)da 积分(从0到2cosa)(4－r^2)rdr
＝积分(从0到pi/2) (8cos^2a－4cos^4a)da
＝2pi－3pi/4
＝5pi/4。