Washers for Revolution Around Shifted Lines (8.12.1) | AP Calculus AB Notes

AP Syllabus focus:
‘For revolution around lines such as y = k or x = c, ring-shaped cross sections still appear, but radii are measured as distances from curves to the chosen line.’

Solids formed by revolving regions around horizontal or vertical lines other than the axes require careful attention to how radii are defined and measured. This subsubtopic develops essential skills for identifying washer-shaped cross sections and setting up definite integrals when the axis of revolution is shifted away from the coordinate axes.

Understanding Washers for Shifted Lines

When a plane region is revolved around a shifted horizontal or vertical line, the resulting solid typically contains a central hole, creating a washer instead of a disc. A washer is a circular ring whose shape arises when the region being revolved does not directly touch the axis of revolution. Identifying a washer requires understanding the structure of the region and how its boundaries relate to the chosen axis.

A plane region in the $xy$ -plane is shown along with a horizontal axis of rotation; when this region is revolved, it generates a solid of revolution. The picture emphasizes how the position of the region relative to the axis determines the cross-sectional shape, including when washers rather than solid disks appear. The specific curves are not important for AP Calculus AB and may include more detail than is required for this subsubtopic. Source.

Outer radius and inner radius are fundamental to constructing washer integrals. The outer radius is the distance from the axis of revolution to the farther curve of the region, while the inner radius is the distance to the closer curve. These distances depend on whether the revolution occurs around a horizontal line, such as $y = k$ , or a vertical line, such as $x = c$ .

Outer Radius: The distance from the axis of revolution to the boundary curve farthest from that axis.

A washer’s shape emerges because the region does not collapse directly onto the axis. This creates an annular cross section with a measurable inner radius.

Inner Radius: The distance from the axis of revolution to the boundary curve closest to that axis.

When the axis of revolution is shifted, determining both radii becomes a matter of subtracting function values or variable coordinates from a constant that represents the new axis.

Measuring Radii When Revolving Around Horizontal Lines

When a region is revolved around a line of the form $y = k$ , radii are computed using vertical distances. Because vertical distance is the difference between $y$ -values, the outer and inner radii take forms such as $|f(x) - k|$ or $|g(x) - k|$ . Choosing which function represents the outer or inner radius depends on which curve lies farther from the axis for all $x$ in the interval.

Common situations include:

Revolving between two curves $y = f(x)$ and $y = g(x)$ around a line above both curves.
Revolving around a line below both curves.
Revolving where the line lies between the curves, forming a solid with an inner void that depends on both functions.

Measuring Radii When Revolving Around Vertical Lines

For a revolution around a line of the form $x = c$ , radii require horizontal distances. Horizontal distance is computed using expressions like $|c - x|$ or $|c - h(y)|$ depending on whether the functions are expressed in terms of $x$ or $y$ .

In AP Calculus AB, students generally maintain functions in $y = f(x)$ form unless geometric or regional considerations make $x = g(y)$ more natural. The decision must ensure that the integration variable aligns with the orientation of the washer slices.

Axis of Revolution: A horizontal or vertical line about which a region is rotated to create a three-dimensional solid.

When the axis shifts, every radius is measured relative to this line rather than the coordinate axes, which changes the structure of the integral while preserving the washer method’s conceptual foundation.

Washer Method Formula Adapted for Shifted Axes

The washer method always requires outer and inner radii, regardless of the axis location. Revolving around shifted lines does not modify the conceptual formula, only the computation of appropriate radii.

$V = \int_{a}^{b} \pi \left[(R(x))^{2} - (r(x))^{2}\right] dx$
$R(x)$ = Outer radius, measured as distance from the far curve to the axis
$r(x)$ = Inner radius, measured as distance from the near curve to the axis

This integral summarizes the accumulation of washer areas across the interval of revolution.

The region that generates a solid of revolution is approximated by a stack of thin rectangles, each of which becomes a thin washer when revolved around the axis. The diagram supports the interpretation of $V = \int \pi\big[(R)^2 - (r)^2\big]$ as the limit of a sum of washer volumes. The specific interpolation and plotting details shown in the original figure go beyond AP Calculus AB requirements and can be ignored by students. Source.

Washers may also be integrated with respect to $y$ when vertical slices are more natural for the geometry of the region. This occurs when equations are given as $x$ in terms of $y$ , or when horizontal washers provide a cleaner representation of the region being revolved.

$V = \int_{c}^{d} \pi \left[(R(y))^{2} - (r(y))^{2}\right] dy$
$R(y)$ = Outer radius expressed using functions of $y$
$r(y)$ = Inner radius expressed using functions of $y$

Choosing $x$ or $y$ as the variable of integration must simplify the structure of the radii, align with the geometry of the region, and ensure a single integral is sufficient.

Process for Setting Up Washer Integrals with Shifted Axes

To analyze solids formed via revolution around a shifted line, students follow a structured approach that ensures each part of the method is logically supported and mathematically sound:

Identify the line of revolution and classify it as horizontal ( $y = k$ ) or vertical ( $x = c$ ).
Determine whether slices perpendicular to the axis of revolution generate washers.
Decide on integration with respect to $x$ or $y$ based on the orientation of the washers.
Compute outer radius by measuring the distance from the farther curve to the axis.
Compute inner radius by measuring the distance from the nearer curve to the axis.
Confirm that the chosen radii remain valid over the entire interval.
Write the corresponding washer integral using the appropriate limits and radii expressions.

This systematic procedure ensures that washer problems involving shifted lines are handled with clarity and consistency, enabling students to interpret how cross-sectional geometry interacts with displaced axes of revolution.

FAQ

Always subtract the coordinate of the axis of rotation from the coordinate of the curve, but the order depends on which quantity is farther from the axis. The radius must always be positive, so choose the order that yields a positive value without requiring absolute values.

If you are unsure, sketch the axis and identify visually which value lies above, below, left, or right; this makes the correct subtraction order clearer.

When the axis lies between curves, each curve must be measured separately from the axis, and the curve nearer to the axis becomes the inner radius.

This often creates radii that change roles depending on the interval, which may require splitting the integral if the region’s geometry changes.

Yes, but only if part of the region touches the axis of rotation despite the axis being shifted. In such cases, the inner radius becomes zero at points of contact.

This tends to occur when the shifted line intersects or grazes the region’s boundary, producing a mixture of discs and washers.

Stretches change the distance between the curve and the axis, altering both radii.

A horizontal stretch modifies how rapidly the curve moves away from the axis left–right, while a vertical stretch alters its height relative to horizontal lines.

Because radii depend directly on these distances, stretched functions typically produce larger or smaller washers across the interval.

When revolving around x = c, washers are perpendicular to the axis, so they are horizontal. Horizontal washers naturally align with y-values.

If the region’s boundaries are easier to express as x in terms of y, integrating with respect to y:

• avoids complex inverses
• ensures radii are simple horizontal distances
• reduces algebraic errors in setting up outer and inner radii

Practice Questions

A region in the plane is bounded above by the curve y = 5 − x and below by the curve y = 1. The region is revolved about the line y = −2. Write an integral expression for the volume of the resulting solid using the washer method.

Question 1 (1–3 marks)

• 1 mark: Correct identification of the outer radius as (5 − x) − (−2) or equivalent.
• 1 mark: Correct identification of the inner radius as 1 − (−2) or equivalent.
• 1 mark: Correct integral of the form integral from x = a to b of pi[(outer radius)^2 − (inner radius)^2] dx.

The region enclosed by the curves y = x + 1 and y = 4, for x between 0 and 2, is revolved about the vertical line x = −1.

(a) Identify the outer and inner radii in terms of y or x, as appropriate.
(b) Set up a definite integral for the volume of the solid formed.
(c) Explain why the washer method, rather than the disc method, is required for this problem.

Question 2 (4–6 marks)

(a)
• 1 mark: Correct identification of the outer radius as the distance from x = −1 to the line y = 4 interpreted in x-form, or correct expression when converted to functions of y.
• 1 mark: Correct identification of the inner radius as the distance from x = −1 to the curve y = x + 1 expressed in the appropriate variable.

(b)
• 1 mark: Correct choice of variable of integration matching the radii (usually integrating with respect to y).
• 1 mark: Correct definite integral using pi[(outer radius)^2 − (inner radius)^2].

(c)
• 1 mark: Correct explanation that the washer method is necessary because the region does not touch the axis of rotation, producing a hollow interior rather than a solid disc.

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Written by:

Dr Rahil Sachak-Patwa

Oxford University - PhD Mathematics

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.