Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y = 3x^5, y = 3x, x ≥ 0; about the x-axis

Question

Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line.

y = 3x⁵,  y = 3x,  x ≥ 0;    about the x-axis

Answer 1

let f(x) = 3x and g(x) = 3x⁵

Intersection: 3x=3x⁵ --> x = -1,0,1 but only want x≥0

V = π∫[f(x)]² - [g(x)]²dx For washer method

V= π∫[3x]²-[3x⁵]²dx = 9π∫x²-x¹⁰]dx integrated from 0 to 1

= 9π[x³/3-x¹¹/11] evaluated from 0 to 1

= 9π[1/3-1/11 -0+0] = 9π(11-3)/33 = 24π/11