find volume using shells method

Question

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about x = 3.

y = 5x⁴,    y = 0,    x = 2

 

Answer 1

The volume of a cylindrical shell is 2πrht where r=radius=3-x, h=height=y, t=thickness=dx.

The integral is 2π∫(3-x)5x⁴dx for 0≤x≤2.

2π∫(15x⁴-5x⁵)dx=2π[3x⁵-5x⁶/6]₀²=2π(96-320/6)=256π/3=268.0826 cubic units approx is the volume of the hollow solid.