Finding Volume by Integration: Rotating Curves About the y-Axis

[JJ99]

Find the volume of the solid formed when the region bounded by the curves y=x³, y=1, and x=0 is rotated about the y-axis, use washer AND shell methods

Disc/Washer:
Ωx²dy = Ω(y^1/3)²dy = Ωy^2/3dy
Volume:
V = 0∫1 Ωy^2/3dy = Ω{(3y^5/3)/5} = 3Ω/5


Shell:
2Ωy(x)dx = 2Ωx(x³)dx = 2Ωx⁴dx
Volume:
V = 0∫1 2Ωx⁴dx = 2Ω{(x⁵)/5} = 2Ω/5


Obviously this question seems simple enough but I'm finding different answers so I'm going wrong somewhere. Any help is appreciated

 

[LCKurtz]

Find the volume of the solid formed when the region bounded by the curves y=x³, y=1, and x=0 is rotated about the y-axis, use washer AND shell methods

Disc/Washer:
Ωx²dy = Ω(y^1/3)²dy = Ωy^2/3dy
Volume:
V = 0∫1 Ωy^2/3dy = Ω{(3y^5/3)/5} = 3Ω/5


Shell:
2Ωy(x)dx = 2Ωx(x³)dx = 2Ωx⁴dx
Volume:
V = 0∫1 2Ωx⁴dx = 2Ω{(x⁵)/5} = 2Ω/5


Obviously this question seems simple enough but I'm finding different answers so I'm going wrong somewhere. Any help is appreciated

For a shell you want $2\pi x(y_{upper}-y_{lower})$ in the integrand. And use $\pi$ instead of $\Omega$.