用极坐标计算下列二重积分:
(1)I=∫∫D(6-3x-2y)dxdy,其中D:x2+y2R2;
(2)I=∫∫D√(R2-x2-y2)dxdy,其中D:x2+y2≤Rx;
(3)I=∫∫Dsin√(x2+y2)dxdy,其中D:π2≤x2+y2≤4π2;
(4)I=∫∫Dln(1+x2+y2)dxdy,其中D是x2+y2≤1在第一象限的部分.
(1)令x=reosθ,y=rsinθ,则 I=∫∫D(6-3x-2y)dxdy=∫02πdθ∫0Rr(6-3rcosθ-2rsinθ)dr =∫02π[(3r2-r3cosθ-(2/3)r3sinθ)∣0R]dθ =∫02π[(3R2-R3cosθ-(2/3)R3sinθ)∣dθ=6πR2 (2)令x=rcosπ,y=rsinθ,则: I=∫-(π/2)2πdθ∫0Rcosθ√(R2-r2)rdr= ∫-(π/2)2π[-(1/2)]dθ∫0Rcosθ√(R2-r2)d(R2-r2) =-(1/4)∫-(π/2)2π[1/√(R2-r2)∣0Rcosθ]dθ =-(1/4)∫-(π/2)2π(1/Rsinθ-1/R)dθ =R3/9(3π-4) (3)令x=rcosθ,y=rsinθ,则: I=∫02πdθ∫π2πrsinrdr=∫02π [(1/2)r2sinr-∫π2π(r2/2)cosrdr]dθ=-6π2 (4)令x=rcosθ,y=rsinθ,则: I=∫∫Dln(1+x2+y2)dxdy=∫ 0π/2dθ∫01rln(1+r2)dr =π/4(2ln2-1)