计算x^2*ydxdy的二重积分,其中D是由x^2-y^2=1及y=0,y=1所围成的平面区域.

Question

计算x^2*ydxdy的二重积分,其中D是由x^2-y^2=1及y=0,y=1所围成的平面区域.

Answer 1

∫∫_D x²y dxdy
= _D₁ x²y dxdy + ∫_D₂ x²y dxdy
= ∫(0→1) dy ∫(0→-√(1 + y²)) x²y dx + ∫(0→1) dy ∫(0→√(1 + y²) x²y dx
= ∫(0→1) [y · x³/3 |(0→-√(1 + y²)) + y · x³/3 |(0→√(1 + y²))] dy
= (1/3)∫(0→1) [- y(1 + y²)^(3/2) + y(1 + y²)^(3/2)] dy
= 0