Volumes of Revolution: Cylindrical Shells

Shells from vertical strips

Choosing radius and height

Agreement as verification

Pappus's theorem preview

A vertical strip at position x of height h(x) and width dx rotates about the y axis. What is the volume of the resulting cylindrical shell?

For which situation does the shell method have a clear advantage over washers?

A region rotates about the vertical line x = a with a to the left of the region. What is the shell radius for a strip at position x?

The region between y = x and y = x^2 on [0, 1] rotates about the y axis. What is the shell height?

The torus from rotating a disk of radius r about an axis at distance R from its center has volume 2 pi squared R r squared. Which factor is the tube's cross-sectional area?

The region under y = sqrt(x) on [0, 1] rotated about the y axis has volume 4 pi/5 by shells. A washer computation must give the same value. Which washer integral is correct?

The region under y = sqrt(x) on [0, 1] rotated about the y axis gives V = 2 pi integral of x^(3/2) dx. What value results, as a fraction of pi? Answer like 4/5.

For a vertical strip at position x rotating about the y axis, the shell radius equals x. At x = 3, what is the radius? Answer with a number.

The shell volume element 2 pi x h(x) dx begins with a constant from the circumference's formula. What number is that constant? Answer with a number.

In the shell formula V = 2 pi integral of x h(x) dx, the 2 pi sits outside the integral. As a single number, how many pi's worth of circumference structure does it carry? Answer with a number.