Volume of Revolution - Shell Method

Calculus 1 · Axiom Academy

EXAMPLE Volume of Revolution — Shell Method Rotate the line y=x about the y -axis and find the volume with cylindrical shells The region bounded by y=x , the x -axis, and the line x=2 is rotated about the y -axis. Find the volume of the resulting solid using the shell method . The region and a representative shell Each thin vertical strip at position x sweeps out a cylindrical shell of radius x and height f(x)=x as the region turns about the y -axis. Nice work — you turned a spinning region into a volume by adding up cylindrical shells. The pieces worth keeping: Shell formula: — each shell's area is (its circumference) times its height. Read off radius and height: rotating about the y -axis with vertical strips, the radius is the distance to the axis, x , and the height is the function value, f(x)=x . Why shells here: the strips are parallel to the axis of rotation, so they sweep into shells — no need to solve y=x for x or split the region. The same solid by the washer method gives too — the methods always agree; you pick whichever keeps the setup clean.

This is the written version of the interactive lesson above. See the full Calculus 1 course.