设x=rcosθ,y=rsinθ (极坐标变换)
则 原式=∫(0,2π)dθ∫(0,R)(r²cos²θ/2+r²sin²θ/3)rdr
=∫(0,2π)(cos²θ/2+sin²θ/3)dθ∫(0,R)r³dr
=R^4/96∫(0,2π)(10+2cos(2θ))dθ
=(R^4/96)*20π
=5πR^4/24
∵πR^4/4(1/2+1/3)=(πR^4/4)*(5/6)=5πR^4/24
∴原式=πR^4/4(1/2+1/3)
故应该选择答案（D）.
设x=rcosθ,y=rsinθ (极坐标变换)
则 原式=∫(0,2π)dθ∫(0,R)(r²cos²θ/2+r²sin²θ/3)rdr
=∫(0,2π)(cos²θ/2+sin²θ/3)dθ∫(0,R)r³dr
=R^4/96∫(0,2π)(10+2cos(2θ))dθ
=(R^4/96)*20π
=5πR^4/24
∵πR^4/4(1/2+1/3)=(πR^4/4)*(5/6)=5πR^4/24
∴原式=πR^4/4(1/2+1/3)
故应该选择答案（D）.