Answer:
[tex]V=\frac{448\pi}{5}[/tex]
Step-by-step explanation:
We are given that curves y=[tex]3x^4[/tex] 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=[tex]3x^4[/tex]
x=0
Hence, x changes from 0 to 2.
Radius =4-x
Height of cylinder =y=[tex]3x^4[/tex]
Surface area of cylinder =[tex]2\pi r h[/tex]
Volume V generated by the rotating curves
=[tex]2\pi\int_{0}^{2} (4-x)(3x^4)dx[/tex]
V=[tex]2\pi\int_{0}^{2}(12x^4-3x^5)dx[/tex]
V=[tex]2\pi[\frac{12x^5}{5}-\frac{x^6}{2}]^2_0[/tex]
V=[tex]2\pi[\frac{384}{5}-32][/tex]
V=[tex]2\pi\frac{384-160}{5}[/tex]
[tex]V=\frac{448\pi}{5}[/tex]
Hence, the volume V generated by rotating the region by the given curves about x =4=[tex]\frac{448\pi}{5}[/tex].
