Calculus II ep05: Volume with washers (Sep 17, 2026)

Overview

Prof Staecker reviews washer volumes with rotation axes away from the x-axis, emphasizing how to identify inner and outer radii as vertical distances from the axis to the region. He then derives the cylindrical-shell formula, V = 2π∫ₐᵇ x f(x) dx for rotation around the y-axis, and compares the methods through examples involving x², √x, and sin x.

Key takeaways

For washer problems with horizontal rotation axes, compute each radius as the upper y-coordinate minus the lower y-coordinate, measured from the axis to the relevant boundary.
The washer method uses V = π∫(R² − r²) dx, so expanding the squared radii can simplify the integrand when substitution lacks a matching derivative.
For shells rotated around the y-axis, radius x, height f(x), and thickness dx combine to give V = 2π∫ₐᵇ x f(x) dx.
Shell integration bounds run from the axis outward; for a symmetric region spanning x = −1 to 1, shells may use only 0 to 1 because each shell already wraps around the axis.
Method choice depends on the resulting integral: washers simplify the √x example to π∫₀¹(1 − x) dx, while shells turn a parabola into the simple integral 2π∫₀¹x³ dx.
Rotating the positive arch of sin x around the y-axis produces 2π∫₀π x sin(x) dx, an example where integration by parts is needed.

Chapters

0:00 Washer Volumes: Outer Radius Squared Minus Inner Radius Squared

The washer formula is V = π∫(R² − r²) dx, where R is the distance to the farther boundary and r is the distance to the nearer boundary.
Subtracting the inner disk removes the hollow space from each circular cross-section.
Prof Staecker contrasts straightforward x-axis examples with problems where the rotation axis is a different horizontal line.

3:00 Finding Washer Radii from a Rotation Axis at y = 1

For a region involving y = x² rotated around y = 1, identify each radius as a vertical distance from y = 1 to a boundary of the region.
The inner radius in the worked setup is 1, while the outer radius is written as 1 + x².
For vertical distances, subtract the lower y-coordinate from the upper y-coordinate rather than relying only on the sketch.

9:00 Expanding and Integrating the Washer Setup

Prof Staecker sets up the example as π∫₁²[(1 + x²)² − 1²] dx.
A direct substitution for 1 + x² does not fit because its derivative, 2x, is absent; expanding the square is more practical.
Expansion simplifies the integrand to 2x² + x⁴, with antiderivative (2/3)x³ + (1/5)x⁵.

13:00 Washer Radii for y = x² Rotated Around y = 1

For the region from x = 0 to x = 1, the rotation axis is y = 1 and the region lies between y = 0 and y = x².
The inner radius is the constant distance 1 from y = 1 to the x-axis.
The outer radius is 1 + x², the distance from y = 1 to y = −x² in the illustrated setup.

16:25 Practice: Four Washer-Radius Setups with Horizontal Axes

Students practice identifying inner and outer radii for four sketched regions, with rotation axes including y = 3 and the x-axis.
For each vertical radius, the reliable procedure is to subtract the lower boundary’s y-value from the upper boundary’s y-value.
The final practice problem uses y = √x over x = 0 to 1, rotated around the x-axis.

24:10 Checking the Four Radius Pairs and Computing the √x Volume

The answer review gives radius pairs including inner 3 − x² and outer 3, inner 2 and outer 1 + x², and inner 2 and outer 2 + x³.
For y = √x rotated around the x-axis, the outer radius is 1 and the inner radius is √x.
The washer integral is π∫₀¹(1² − (√x)²) dx = π∫₀¹(1 − x) dx = π/2.

28:25 Cylindrical Shells: Slicing a Rotated Shape into Rings

Prof Staecker introduces cylindrical shells as an alternative to slicing a solid into washers.
For a region rotated around the y-axis, a vertical rectangular slice sweeps out a thin cylindrical shell.
Shells can be harder to visualize than washers, but they can make some integrals simpler.

32:00 Unrolling a Shell to Derive Its Volume

A thin shell has height f(x), thickness Δx, and radius x when the axis of rotation is the y-axis.
Unrolling the shell produces a rectangle whose length is the circumference 2πx and whose height is f(x).
The thin-shell volume is approximately 2πx f(x) Δx; as Δx becomes dx, summing the shells gives an integral.

38:10 The Shell Formula and a Parabola Rotated Around the y-Axis

For rotation around the y-axis, the shell formula is V = 2π∫ₐᵇ x f(x) dx; unlike the washer formula, it does not square the radii.
The bounds measure shells from the axis outward, so the region y = x² from x = 0 to 1 uses bounds 0 and 1 rather than −1 and 1.
For this parabola, V = 2π∫₀¹ x·x² dx = π/2.

46:20 Shells for sin x and Choosing a Method by the Integrand

Rotating the first positive arch of y = sin x around the y-axis gives the setup V = 2π∫₀π x sin(x) dx.
The product x sin(x) requires integration by parts, which Prof Staecker identifies as a technique covered later in the course.
The shell radius represents distance to the rotation axis and the shell height represents the region’s vertical height; both may need adjustment when the axis changes.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Prof Staecker.