求积分:∫∫xy^3do,其中D(,∫∫下面有个D)是是由y^2=4x,y=x-1所围成的闭区域

Question

求积分:∫∫xy^3do,其中D(,∫∫下面有个D)是是由y^2=4x,y=x-1所围成的闭区域

Answer 1

D ＝ ｛x = y²/4 → y+1,y = 2 - 2√2→2 + 2√2｝
∫∫xy³ dxdy
= ∫y³ x² / 2 dy x = y²/4 → y+1
= 1/2∫y³ [ (y+1)²-y⁴/16] dy
= 1/32 ∫y³ [ 16(y+1)²-y⁴] dy
= 1/32 ∫y³ [ -y⁴+ 16y² + 32y +16] dy
= 1/32 ∫ [ -y^7+ 16y^5 + 32y^4 +16y³ ] dy y = 2 - 2√2→2 + 2√2
= 1/32 [ -1/8y^8+ 16/6y^6 +32/4y^5+16/4y^4 ] dy y = 2 - 2√2→2 + 2√2
≈ 495