Question

Question Part Points Submissions Used Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about x = 4. y = 3x4, y = 0, x = 2.

Answer 1

`V=(448\pi)/(5)`

Explanation:

We are given that curves y=
`3x^4` is rotated about x=4 .

Given that y=0 and x=2

We have to find the volume V generated by rotating the region bounded by the curves with the help of method of cylindrical shells.

First we find the intersection point

Substitute y=0 then we get

0=
`3x^4`

x=0

Hence, x changes from 0 to 2.

Radius =4-x

Height of cylinder =y=
`3x^4`

Surface area of cylinder =
`2\pi r h`

Volume V generated by the rotating curves

=
`2\pi\int_(0)^(2) (4-x)(3x^4)dx`

V=
`2\pi\int_(0)^(2)(12x^4-3x^5)dx`

V=
`2\pi[(12x^5)/(5)-(x^6)/(2)]^2_0`

V=
`2\pi[(384)/(5)-32]`

V=
`2\pi(384-160)/(5)`

`V=(448\pi)/(5)`

Hence, the volume V generated by rotating the region by the given curves about x =4=
`(448\pi)/(5)`.