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

Answer:

$V=\frac{448\pi}{5}$

Step-by-step 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=$2\pi[\frac{12x^5}{5}-\frac{x^6}{2}]^2_0$

V=$2\pi[\frac{384}{5}-32]$

V=$2\pi\frac{384-160}{5}$

$V=\frac{448\pi}{5}$

Hence, the volume V generated by rotating the region by the given curves about x =4=$\frac{448\pi}{5}$.