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:[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=2We 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=0Hence, x changes from 0 to 2.Radius =4-xHeight 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].