Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the curves 𝑦=𝑥3,y=x3, 𝑦=8,y=8, and 𝑥=0x=0 about the 𝑥x-axis.
See the graph for this region : https://www.desmos.com/calculator/q9jn22a8q4
Because we are rotating about the x axis, the area will be a function of y.....so.... we need to write y =x^3 as x = y^1/3
So we have
8
2pi ∫ radius * height dy =
0
8
2pi ∫ y * y^(1/3) dy =
0
8
2pi ∫ y ^ (4/3) dy =
0
8
2pi * [ ( 3/7) y^(7/3) ] =
0
(6 pi / 7 ) * [ ( 8^(7/3) - 0^(7/3) ] =
(6 pi /7) * [ 8^(1/3) }^7 =
(6pi /7) * [ 2^7 ] =
(6 pi / 7) * 128 ≈ 344.68 units^3