计算二重积分D∫∫(x+6y)dσ,其中D由直线y=x,y=5x,x=1 ,是由 所围成的区域.

X型：
∫∫D (x + 6y) dσ
= ∫(0→1) dx ∫(x→5x) (x + 6y) dy
= ∫(0→1) [∫(x→5x) (x + 6y) dy] dx
= ∫(0→1) [xy + 3y² |(y = x→5x)] dx
= ∫(0→1) [x(5x) + 3(5x)²] - [x(x) + 3(x)²] dx
= ∫(0→1) 76x² dx
= 76x³/3 |(x = 0→1)
= 76/3
Y型：
∫∫D (x + 6y) dσ
= ∫(0→1) dy ∫(y/5→y) (x + 6y) dx + ∫(1→5) dy ∫(y/5→1) (x + 6y) dx
= ∫(0→1) [∫(y/5→y) (x + 6y) dx] dy + ∫(1→5) [∫(y/5→1) (x + 6y) dx] dy
= ∫(0→1) [x²/2 + 6xy |(x = y/5→y)] dy + ∫(1→5) [x²/2 + 6xy |(x = y/5→1)] dy
= ∫(0→1) [(y²/2 + 6y²) - ((y/5)²/2 + 6y²/5)] dy + ∫(1→5) [(1/2 + 6y) - ((y/5)²/2 + 6y²/5)] dy
= ∫(0→1) 132y²/25 dy + ∫(1→5) (- 61y²/50 + 6y + 1/2) dy
= 76/3