Use the disk method or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about each given line.

Question

y = x³

y = 0

x = 2

(a) the x-axis

(b) the y-axis

(c) the line x = 9

Answer 1

(a) About x-axis: V = π∫₀²(x³)²dx = πx⁷/7₀² = 128π/7

(b) x = 2, then y = 2³ = 8; x = y^1/3

V = π∫₀⁸(2² - y^2/3)dy = π (4y - 3y^5/3/5)₀⁸ = π(32 - 96/5) = 64π/5

(c) V = π∫₀⁸[(9 - y^1/3)² - 7²]dy = π∫₀⁸(32 - 18y^1/3 + y^2/3)dy = π(32y - 27/2y^4/3 + 3/5y^5/3)₀⁸ =

π(256 - 216 + 96/5) = 296π/5