# Finding volume by integration

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(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

## Answers and Replies

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##.