# Finding volume by integration

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(y1/3)2dy = Ωy2/3dy
Volume:
V = 0∫1 Ωy2/3dy = Ω{(3y5/3)/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
Homework Helper
Gold Member
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(y1/3)2dy = Ωy2/3dy
Volume:
V = 0∫1 Ωy2/3dy = Ω{(3y5/3)/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##.