Question

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis. y = 2x³, y = 2x, for x ≥ 0

Answer 1

`\displaystyle A=(8\pi)/(15)`

Explanation:

Shell Method (Vertical Axis)

`\displaystyle A=2\pi\int^b_ar(x)h(x)dx`

Radius:
`r(x)=y`

Height:
`h(x)=2x-2x^3`

Bounds:
`[a,b]=[0,1]`

Set up integral and evaluate

`\displaystyle A=2\pi\int^1_0x(2x-2x^3)dx\\\\A=2\pi\int^1_0(2x^2-2x^4)dx\\\\A=2\pi\biggr((2)/(3)x^3-(2)/(5)x^5\biggr)\biggr|^1_0\\\\A=2\pi\biggr((2)/(3)(1)^2-(2)/(5)(1)^2\biggr)\\\\A=2\pi\biggr((2)/(3)-(2)/(5)\biggr)\\\\A=2\pi\biggr((10)/(15)-(6)/(15)\biggr)\\\\A=2\pi\biggr((4)/(15)\biggr)\\\\A=(8\pi)/(15)`