Find the volume of the solid revolved around y=x^3, y=1, x=2 about the x-axis

Answer 1

Draw the region. It is bounded above by the graph of y = x³ and below by the horizontal line y = 1. The x-coordinates of all points in the region lie in the interval [1,2].

A thin vertical cross section at an average distance of x from the y-axis rotated about the x-axis generates a washer with outer radius x³ and inner radius 1.

V = ∫_{(from 1 to 2)} [π(x³)² - π(1)²]dx = π[(1/7)x⁷ - x) _{(from 1 to 2)}