Washer integrals for horizontal or vertical lines allow volumes of solids of revolution to be computed by measuring changing radii relative to shifted axes using definite integrals.

Washer Integrals for Horizontal or Vertical Lines

A washer-based volume arises when a region is revolved around a horizontal or vertical line, producing a solid with a hollow center. This hollow center creates a ring-shaped cross section whose area must be computed by comparing an outer radius to an inner radius. Because these radii depend on the distance from the region’s bounding curves to the chosen line of revolution, setting up the correct expressions is essential for accurate volume calculations. In this subsubtopic, the AP Calculus AB focus is on expressing these radii as differences between functions and the axis of revolution, then writing the corresponding integral using the washer formula.

Understanding Horizontal and Vertical Lines of Revolution

When revolving a region around a horizontal line such as $y = k$, the radii measure vertical distances between the curve(s) and the axis. When revolving around a vertical line such as $x = c$, the radii measure horizontal distances. The orientation determines whether the integral must be written with respect to $x$ or $y$, based on how the radius naturally varies across the region.

Key Idea: Measuring Distance to the Axis

The radius of revolution is always defined as a nonnegative distance from a curve to the axis being used. Distances depend on whether the curve lies above or below a horizontal axis or to the right or left of a vertical axis.

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The integral sums the volumes of these infinitesimally thin washers to give the total volume of the solid of revolution.

A planar region is sliced by a thin rectangle perpendicular to a horizontal axis of revolution. When the slice is revolved, it generates a washer-shaped cross-section whose outer and inner edges track the bounding curves. This image emphasizes how a continuous family of such washers accumulates to form the full solid. Source.

Determining Whether to Integrate with Respect to x or y

Choosing the correct variable prevents unnecessary complications and ensures the washer method is applied efficiently. The selection depends on the orientation of the axis:

• Revolving around a horizontal line ($y = k$)

  • Radii are vertical distances.

  • Use integration with respect to $x$, provided the curves are functions of $x$.

• Revolving around a vertical line ($x = c$)

  • Radii are horizontal distances.

  • Use integration with respect to $y$, provided the curves are functions of $y$.

If the region has boundaries more naturally expressed in the opposite orientation, re-expressing the curves may simplify the radii expressions.

Constructing the Outer and Inner Radii

The washer method always requires identifying two distances:

• Outer radius: distance from the most distant bounding curve to the axis

• Inner radius: distance from the closest bounding curve to the axis

These distances must always remain nonnegative and must correctly reflect which curve is farther or nearer across the entire interval.

When the Axis Is Horizontal ($y = k$)

To compute radii:

• Outer radius:

  • If the upper curve is $y = f(x)$, then

    • distance = $|f(x) - k|$.

• Inner radius:

  • If the lower curve is $y = g(x)$$, then</p><ul><li><p>distance =$|g(x) - k|$.</p></li></ul></li></ul></li></ul><p>Because the geometry usually ensures one curve stays above the other on the interval, the absolute value may be replaced with direct subtraction in the correct order. Still, conceptualizing it as a distance helps students avoid errors.</p><p>For a region rotated around a horizontal line$y=k$, the outer radius is the vertical distance from$y=k$to the top function and the inner radius is the distance to the bottom function.</p><img src="https://tutorchase-production.s3.eu-west-2.amazonaws.com/0a47abb2-efb0-49c7-b5cb-a0d4a3f780e6-file.png" alt="Pasted image" style="max-width: 100$x = c$)</strong></h3><p>To compute radii:</p><ul><li><p><strong>Outer radius:</strong></p><ul><li><p>If the rightmost curve is$x = f(y)$, then</p><ul><li><p>distance =$|f(y) - c|$.</p></li></ul></li></ul></li><li><p><strong>Inner radius:</strong></p><ul><li><p>If the leftmost curve is$x = g(y)$, then</p><ul><li><p>distance =$|g(y) - c|$.</p></li></ul></li></ul></li></ul><p>Working with functions of$y$often provides a simpler setup in vertical-axis problems, especially when the region has top and bottom boundaries that are difficult to express in terms of$x$.</p><h2 class="editor-heading" id="structural-steps-for-setting-up-washer-integrals"><strong>Structural Steps for Setting Up Washer Integrals</strong></h2><p>Students benefit from a clear procedure that organizes the geometry and algebra involved. The syllabus emphasizes interpreting graphs and expressing radii using correct distance formulas.</p><h3 class="editor-heading"><strong>Process for Any Washer Method Problem</strong></h3><ul><li><p><strong>Identify the line of revolution.</strong></p><ul><li><p>Determine whether it is horizontal or vertical.</p></li></ul></li><li><p><strong>Label the outer and inner curves.</strong></p><ul><li><p>Decide which curve is farther from the axis on each slice.</p></li></ul></li><li><p><strong>Express radii as distances.</strong></p><ul><li><p>Use absolute difference between function and axis until the correct order is confirmed.</p></li></ul></li><li><p><strong>Determine the variable of integration.</strong></p><ul><li><p>Match vertical slices to$x$-integration and horizontal slices to$y$-integration.</p></li></ul></li><li><p><strong>Write the washer integral.</strong></p><ul><li><p>Use$\pi[(\text{outer radius})^{2} - (\text{inner radius})^{2}]$ with appropriate limits.

• Ensure radii remain nonnegative over the entire interval.

Visual Reasoning with Axes of Revolution

Conceptualizing the geometry strengthens understanding. A horizontal line of revolution produces washers stacked left to right, while a vertical line of revolution produces washers stacked bottom to top. Because the shape of each washer depends on distance to the axis, graph inspection and boundary verification prevent sign errors in radius expressions.