Washer Method: Other Axes — Volume with Washer Method: Revolving Around Other Axes - Calculus 1

SECTION 1

Historical Context & Motivation

The problem of computing volumes of curved solids has fascinated mathematicians for over two thousand years. Ancient Greek mathematicians developed ingenious geometric arguments to find the volumes of spheres, cones, and other shapes formed by rotating curves. The washer method is a modern calculus technique that traces its roots directly to these early efforts, extending them far beyond what the Greeks could have imagined.

~250 BCE • Archimedes and the Method of Exhaustion

Archimedes calculated the volume of a sphere by approximating it with stacks of thin cylindrical slices—an early precursor to integration and the disk/washer approach.

1615 • Kepler's Wine Barrel Problem

Johannes Kepler investigated the volumes of barrels (solids of revolution) to determine the optimal barrel shape, inspiring new methods for computing rotational volumes.

1635 • Cavalieri's Principle

Bonaventura Cavalieri formalized the idea that solids with equal cross-sectional areas at every height have equal volumes, laying the groundwork for slicing methods in calculus.

1660s • Newton & Leibniz Develop Calculus

The invention of integral calculus gave mathematicians a systematic tool—the definite integral—to sum infinitely many infinitesimally thin slices, making the washer method possible in its modern form.

The standard washer method typically revolves a region around the x-axis or y-axis. But what happens when the axis of rotation is a different horizontal or vertical line, such as y = 5 or x = −2? The distances from each curve to the axis change, and the formulas for the outer and inner radii must be adjusted. This lesson answers the question: how do we set up and evaluate a washer integral when the axis of revolution is not one of the coordinate axes?

SECTION 2

Core Principles & Definitions

Before diving into calculations, you need to solidify a few foundational ideas. The washer method is an extension of the disk method, and understanding the relationship between the two is essential. When a region between two curves is revolved around an axis, each cross-section perpendicular to the axis looks like a washer—a disk with a hole cut out of its center.

Solid of Revolution

A three-dimensional solid formed by rotating a two-dimensional region around a fixed line (the axis of revolution). The shape of the solid depends on both the region and which axis you choose.

Outer Radius (R)

The distance from the axis of revolution to the curve that is farther from the axis. This defines the outer edge of each washer.

Inner Radius (r)

The distance from the axis of revolution to the curve that is closer to the axis. This defines the hole in each washer.

Adjusting for Other Axes

When the axis is not y = 0 or x = 0, you compute each radius as the signed distance between the curve's value and the axis line, using subtraction: |f(x) − axis value|.

Integration Variable

If the axis is horizontal (e.g., y = k), integrate with respect to x. If the axis is vertical (e.g., x = k), integrate with respect to y. The slicing is always perpendicular to the axis.

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KEY TAKEAWAY

Think of a washer like a DVD or a bagel slice. The outer radius is the distance from the center hole to the outside edge, and the inner radius is the distance from the center hole to where the solid begins. When you move the axis of revolution away from the coordinate axes, it's like moving the spindle of a pottery wheel—the shape of your pot changes because every point is now a different distance from the spindle.

SECTION 3

Visual Explanation

The diagram below shows a region between two curves being revolved around the line y = −1, which lies below the x-axis. Notice how the outer radius R and inner radius r are both measured from the axis y = −1, not from the x-axis. This shift is the key difference when revolving around other axes.

The shaded region between f(x) (cyan) and g(x) (pink) is revolved around the dashed red line y = −1. Both R and r are measured from the axis y = −1, so each radius involves adding 1 to the function value.

In the diagram above, the critical observation is that we are no longer measuring distances from the x-axis. Every radius is the distance from the axis of revolution to the curve. When the axis is y = −1, we subtract −1 from the function value, which is the same as adding 1. If the axis were above the region, say y = 6, the distances would be computed as 6 − f(x) and 6 − g(x), and you'd need to carefully determine which curve is farther from the axis to identify the outer radius.

SECTION 4

Mathematical Framework

The general washer formula remains the same regardless of which axis you rotate around. What changes is how you define the outer and inner radii. Let's build the formulas step by step.

GENERAL WASHER FORMULA

V = π ∫ₐᵇ [R(x)]² − [r(x)]² dx

V = volume of the solid, R(x) = outer radius (farther curve from axis), r(x) = inner radius (closer curve from axis), a and b = bounds of integration along the x-axis.

HORIZONTAL AXIS BELOW REGION: y = k (where k < region)

R(x) = f(x) − k, r(x) = g(x) − k

When the axis y = k is below both curves, the farther curve f(x) gives the outer radius and the closer curve g(x) gives the inner radius. Each radius is found by subtracting the axis value k from the function value.

HORIZONTAL AXIS ABOVE REGION: y = k (where k > region)

R(x) = k − g(x), r(x) = k − f(x)

When the axis y = k is above both curves, the roles reverse: the curve that was 'lower' (g(x)) is now farther from the axis, making it the outer radius. Each radius is k minus the function value.

VERTICAL AXIS (INTEGRATE WITH RESPECT TO y)

V = π ∫_c^d [R(y)]² − [r(y)]² dy

When revolving around a vertical line x = k, express both curves as functions of y. R(y) = distance from axis to the farther curve, r(y) = distance from axis to the closer curve. Integrate from y = c to y = d.

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Common Mistake Alert

Students often forget to check which curve is farther from the axis after the axis shifts. When the axis moves from y = 0 to y = 6 (above the region), the curve that was on top is now closer to the axis—so it becomes the inner radius, not the outer radius. Always sketch the setup and identify which curve is farther from the axis of revolution.

SECTION 5

Setting Up Radii for Different Axes

The trickiest part of the washer method with other axes is correctly expressing the radii. The table and diagram below cover the four most common scenarios you'll encounter. In each case, assume the region is bounded by y = f(x) on top and y = g(x) on the bottom, between x = a and x = b.

Axis of Revolution	Outer Radius R	Inner Radius r	Key Idea
y = k (below region)	f(x) − k	g(x) − k	Subtract axis from each curve
y = k (above region)	k − g(x)	k − f(x)	Subtract each curve from axis; roles swap
x = k (left of region)	f(y) − k (right curve)	g(y) − k (left curve)	Use functions of y; subtract axis value
x = k (right of region)	k − g(y) (left curve)	k − f(y) (right curve)	Subtract curves from axis; roles swap

Summary of radius formulas for four axis positions

Four common cases for axis placement. In Cases 2 and 4, the roles of the outer and inner curves swap because the axis is on the opposite side of the region. Always identify which curve is farther from the axis to determine R.

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REMEMBER THIS RULE

When the axis is on the same side as the 'top' or 'right' curve, that curve becomes the inner radius, not the outer. Always ask yourself: which curve is farther from the axis? That's your outer radius R.

SECTION 6

Worked Example

Let's work through a complete problem: find the volume of the solid formed by revolving the region bounded by y = x² and y = x about the line y = 3.

Volume by Revolving Around y = 3

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Step 1 — Find Intersection Points

Set x² = x. This gives x² − x = 0, so x(x − 1) = 0. The curves intersect at x = 0 and x = 1. These are our bounds of integration: a = 0, b = 1.

Bounds: x = 0 to x = 1
2
Step 2 — Identify Which Curve Is on Top

Between x = 0 and x = 1, we can check x = 0.5: y = x gives 0.5, while y = x² gives 0.25. So y = x is above y = x² in this interval.

Top curve: y = x; Bottom curve: y = x²
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Step 3 — Determine Outer and Inner Radii

The axis y = 3 is above both curves (since both curves have values between 0 and 1 on this interval). When the axis is above the region, the curve farther from the axis is the bottom curve. So R(x) = 3 − x² (distance from axis to bottom curve) and r(x) = 3 − x (distance from axis to top curve).

R(x) = 3 − x², r(x) = 3 − x
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Step 4 — Set Up the Integral

Plug into the washer formula: V = π ∫₀¹ [(3 − x²)² − (3 − x)²] dx. Expand each square: (3 − x²)² = 9 − 6x² + x⁴ and (3 − x)² = 9 − 6x + x².

V = π ∫₀¹ [(9 − 6x² + x⁴) − (9 − 6x + x²)] dx
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Step 5 — Simplify the Integrand

Subtract the two expanded expressions: 9 − 6x² + x⁴ − 9 + 6x − x² = x⁴ − 7x² + 6x. So we have V = π ∫₀¹ (x⁴ − 7x² + 6x) dx.

V = π ∫₀¹ (x⁴ − 7x² + 6x) dx
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Step 6 — Integrate and Evaluate

Integrate term by term: ∫(x⁴ − 7x² + 6x) dx = x⁵/5 − 7x³/3 + 3x². Evaluate from 0 to 1: (1/5 − 7/3 + 3) − (0) = 1/5 − 7/3 + 3. Find a common denominator (15): 3/15 − 35/15 + 45/15 = 13/15.

V = 13π/15 ≈ 2.72 cubic units

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Checking Your Work

Always verify that your outer radius is larger than your inner radius at a test point. At x = 0.5: R = 3 − 0.25 = 2.75 and r = 3 − 0.5 = 2.5. Since R > r, the setup is correct. If you ever get R < r, you've mixed up the radii.

SECTION 7

Washer Method vs. Other Volume Techniques

The washer method is one of several techniques for finding volumes of revolution. Understanding when to use each method will save you time and reduce errors. The table below compares the main approaches.

Method	Best For	Limitation
Disk Method	Solid with no hole (region touches the axis)	Cannot handle a gap between the region and the axis
Washer Method	Solid with a hole; axis may be any horizontal or vertical line	Slicing must be perpendicular to the axis; curves must be expressible as functions of the integration variable
Shell Method	When the natural variable is parallel to the axis (avoids solving for x or y)	Requires integrating in the direction parallel to the axis, which can feel less intuitive at first

Comparison of volume-of-revolution methods

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WHEN TO CHOOSE THE WASHER METHOD

Use the washer method when you can easily express both bounding curves as functions of the variable you're integrating with respect to, and you're slicing perpendicular to the axis of revolution. If expressing curves in terms of the correct variable requires solving messy equations, the shell method may be a better choice.

SECTION 8

Connection to Advanced Topics

Mastering the washer method with non-standard axes prepares you for several advanced ideas in calculus and beyond. The fundamental skill—computing distances from a reference line—appears repeatedly in higher mathematics and applied science.

This Lesson	Advanced Extension
Revolving around horizontal/vertical lines	Pappus' theorem: volume = 2π × (distance of centroid to axis) × area, giving a shortcut for symmetric regions
Computing R(x) and r(x) from axis distance	In multivariable calculus, distances are computed from planes and lines in 3D using similar subtraction formulas
Choosing between washer and shell methods	In differential equations and physics, selecting the right coordinate system (Cartesian vs. polar vs. cylindrical) mirrors this strategic choice
Integrating π[R² − r²]	Engineering applications compute cross-sectional areas of pipes, tubes, and annular structures using the same R² − r² formula

How this lesson connects to future coursework

In Calculus 2 and 3, you'll encounter volumes computed using double and triple integrals, as well as integration in polar and cylindrical coordinates. The ability to set up distances and radii relative to any reference line or surface is a skill that will transfer directly into those more advanced settings.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

The region between y = x and y = x² (from x = 0 to x = 1) is revolved about y = 5. Without computing the integral, explain which curve gives the outer radius and which gives the inner radius. Why do the roles swap compared to revolving around y = 0?

Show answer

When revolving around y = 5 (above the region), the curve farther from the axis is y = x² (the bottom curve), so the outer radius is R = 5 − x². The curve closer to the axis is y = x (the top curve), giving the inner radius r = 5 − x. The roles swap because 'farther from the axis' is now 'farther below y = 5,' which means the lower curve is actually farther away. When the axis was y = 0, the top curve (y = x) was farther, making it the outer radius.

Question 2: BASIC CALCULATION

Find the volume of the solid formed by revolving the region bounded by y = √x and y = 0, from x = 0 to x = 4, about the line y = −2.

Show answer

The axis y = −2 is below the region. The outer radius is R = √x − (−2) = √x + 2, and since the region touches y = 0 at the bottom, the inner radius is r = 0 − (−2) = 2. Set up: V = π ∫₀⁴ [(√x + 2)² − 4] dx = π ∫₀⁴ [x + 4√x + 4 − 4] dx = π ∫₀⁴ [x + 4√x] dx. Integrate: x²/2 + 4 · (2/3)

x^{3/2}

= x²/2 + (8/3)

x^{3/2}

. Evaluate from 0 to 4: 16/2 + (8/3)(8) = 8 + 64/3 = 88/3. So V = 88π/3 ≈ 92.15 cubic units.

Question 3: INTERMEDIATE

Find the volume of the solid generated by revolving the region bounded by y = x² and y = 4 about the line x = 3. (Hint: you'll integrate with respect to y.)

Show answer

The region is bounded on the right by x = √y (from y = x²) and on the left by x = −√y, but since we're in the first quadrant and the region is symmetric, consider x = √y from y = 0 to y = 4. The axis x = 3 is to the right of the region. The outer radius is R = 3 − (−√y) = 3 + √y (if using the full region) or, considering only x ≥ 0: R = 3 − 0 = 3... Actually, let's clarify: the region between y = x² and y = 4 includes both sides. Taking the right half (x = 0 to x = √y): R(y) = 3 − 0 = 3 and r(y) = 3 − √y. So V = π ∫₀⁴ [9 − (3 − √y)²] dy = π ∫₀⁴ [9 − 9 + 6√y − y] dy = π ∫₀⁴ [6√y − y] dy = π[

4y^{3/2}

− y²/2] from 0 to 4 = π[32 − 8] = 24π. By symmetry for the left half the total is doubled if the full parabola is used, but typically this problem considers x ≥ 0, giving V = 24π ≈ 75.40 cubic units.

Question 4: APPLIED

A decorative vase is modeled by revolving the region between y = 2x and y = x² (for 0 ≤ x ≤ 2) around the line y = −1. Find the volume of material needed to make the vase (in terms of π).

Show answer

The axis y = −1 is below both curves. The outer radius is R(x) = 2x − (−1) = 2x + 1, and the inner radius is r(x) = x² − (−1) = x² + 1. Set up: V = π ∫₀² [(2x + 1)² − (x² + 1)²] dx. Expand: (2x + 1)² = 4x² + 4x + 1 and (x² + 1)² = x⁴ + 2x² + 1. Subtract: 4x² + 4x + 1 − x⁴ − 2x² − 1 = −x⁴ + 2x² + 4x. Integrate: −x⁵/5 + 2x³/3 + 2x². Evaluate from 0 to 2: −32/5 + 16/3 + 8 = (−96 + 80 + 120)/15 = 104/15. So V = 104π/15 ≈ 21.78 cubic units.

Question 5: CRITICAL THINKING

Consider the region between y = x and y = x² from x = 0 to x = 1. Let V(k) be the volume obtained by revolving this region about the horizontal line y = k. Without fully computing V(k), determine: for what value of k does V(k) achieve its minimum? Justify your reasoning. (Hint: set up V(k) as an integral and think about what minimizes the integrand.)

Show answer

Set up V(k) using the washer method. If k is above the region: V = π ∫₀¹ [(k − x²)² − (k − x)²] dx. If k is below: V = π ∫₀¹ [(x − k)² − (x² − k)²] dx. In both cases, expanding gives V(k) = π ∫₀¹ [(x)² − (x²)² + 2k(x² − x)] dx when k is below, or V(k) = π ∫₀¹ [(x²)² − (x)² + 2k(x − x²)] dx when k is above... More carefully: for any k, V = π ∫₀¹ |[distance from farther curve]² − [distance from closer curve]²| dx. After careful expansion for arbitrary k, V(k) = π ∫₀¹ (x − x²)(2k − x − x²) dx when the axis is above the centroid, or similar. V(k) simplifies to a quadratic in k, and differentiating with respect to k and setting dV/dk = 0 gives k = ∫₀¹ (x + x²)(x − x²) dx / [2∫₀¹ (x − x²) dx], which equals k = 1/2. The minimum volume occurs at y = 1/2, which is the y-coordinate of the centroid of the region. This connects to Pappus' theorem: the volume is minimized when the axis passes through the centroid, making the 'average' rotation distance smallest.

SUMMARY

Lesson Summary

Summary

The washer method computes the volume of a solid of revolution by integrating π[R(x)² − r(x)²] (or the corresponding expression in y). When the axis of revolution is a line other than the x-axis or y-axis, the outer radius R and inner radius r must be recalculated as distances from the new axis to each bounding curve. If the axis is below or to the left of the region, subtract the axis value from each curve. If the axis is above or to the right, subtract each curve from the axis value.

The critical step is always identifying which curve is farther from the axis — that curve defines the outer radius. When the axis moves to the opposite side of the region, the roles of the two curves swap. Always sketch the region and axis, label R and r, and verify at a test point that R > r before integrating. This technique extends naturally to vertical axes (where you integrate with respect to y) and connects to advanced topics like Pappus' theorem and multivariable integration.