Picture a factory that mass-produces water bottles. Engineers must know the exact volume inside each curved shape so every customer gets the right amount of liquid. Techniques for finding that volume often come straight from AP® Calculus: the disc method vs washer method.

This article has one clear goal: show when and how to use each method. Students will see the formulas, learn a quick decision checklist, and walk through step-by-step examples that mirror the AP® exam style.

Setting the Stage: Solids of Revolution & Definite Integrals

What Is Revolving a Region?

Take a flat region on the coordinate plane and spin it around a line (called the axis of rotation). The flat region sweeps out a 3-D solid.

Visual Snapshot

Imagine the curve y = √x from x = 0 to x = 4. When rotated around the x-axis it looks like a cone that has been smoothed out. A quick sketch should label the axis, show the curve, and mark the bounds.

Disc Method Deep Dive

Core Idea

If the region touches the axis, each cross-section is a solid “pancake” disc. Stack the discs, add their volumes with integration, and the total volume appears.

Formula Box

Vertical slices (function of x):

V = \int_{a}^{b} \pi [R(x)]^{2}dx

Horizontal slices (function of y):

V = \int_{c}^{d} \pi [R(y)]^{2}dy

Picking Slice Direction

• Curve given as y = f(x) and rotation around x-axis → choose vertical slices (dx).
• Curve given as x = g(y) or rotation around y-axis → often easier with horizontal slices (dy).

Example #1: Rotate y = √x, 0 ≤ x ≤ 4 around the x-axis

1. Sketch: Draw the curve from (0, 0) to (4, 2). Because the region touches the x-axis, discs apply.
2. Radius: The distance from the x-axis to the curve is y = √x, so R(x) = √x.
3. Setup:
   • V = \int_{0}^{4} \pi[\sqrt{x}]^{2}dx = \pi\int_{0}^{4} xdx
4. Evaluate:
   • V = \pi\left[\frac{x^{2}}{2}\right]_{0}^{4} = \pi\left(\frac{16}{2}\right)=8\pi cubic units.

Therefore, the bottle-like solid holds 8π units³.

Calculator Tips

• On TI-84 or similar, use fnInt( π*(Y1)^2 , X , 0 , 4 ).
• Remember units. Volume always comes in cubic units.

Washer Method Deep Dive

Why Washers?

If a gap sits between the region and the axis, each cross-section looks like a washer (a disc with a central hole), much like a doughnut.

Formula Box

Vertical slices:

V = \int_{a}^{b} \pi\Big[(R_{\text{outer}})^{2} - (R_{\text{inner}})^{2}\Big]dx

Horizontal slices:

V = \int_{c}^{d} \pi\Big[(R_{\text{outer}})^{2} - (R_{\text{inner}})^{2}\Big]dy

When to Switch

Whenever the region does not touch the axis, choose washers. The inner radius handles the empty space.

Example #2: Region between y = x and y = x², rotated around the x-axis

1. Sketch: The two curves meet at x = 0 and x = 1. y = x sits above y = x².

2. Radii:
   • Outer radius: R(x) = y = x (farther curve).
   • Inner radius: r(x) = y = x² (closer curve).
3. Setup:
   • V = \int_{0}^{1} \pi\big[(x)^{2} - (x^{2})^{2}\big]dx = \pi\int_{0}^{1} \big(x^{2} - x^{4}\big)dx
4. Evaluate:
   • V = \pi\left[\frac{x^{3}}{3} - \frac{x^{5}}{5}\right]_{0}^{1} = \pi\left(\frac{1}{3} - \frac{1}{5}\right) = \pi\left(\frac{2}{15}\right)=\frac{2\pi}{15} cubic units.

Example #3 (Outline Only)

Region between x = y² and x = 2y, rotated around x = 5. Use horizontal washers (dy). Outer radius = 5 – y², inner radius = 5 – 2y. Integrate from y = 0 to y = 4.

Disc Method vs Washer Method: How to Decide

Quick Checklist

• Region touches the axis → discs.
• Visible gap → washers.

Comparison Table

Feature	Disc Method	Washer Method
Cross-section	Solid	Hollow
Formula	\pi R^{2}	\pi(R_{\text{outer}}^{2} - R_{\text{inner}}^{2})
Typical sketch clue	Curve meets axis	Space between curve and axis

Decision Guide

1. Draw the region and the axis.
2. Ask: “Do the slices hit the axis?”
3. If yes, choose discs; otherwise, choose washers.
4. Then decide dx vs dy based on slice direction.

Mini Example #4: Rotate region bounded by y = 2 – x and y = 0 around y = –1

• Sketch shows a triangle under the line from x = 0 to x = 2.
• Gap of 1 unit to y = –1 means washers.
• Radii: R(x) = (2 – x) – (–1) = 3 – x, r(x) = 0 – (–1)=1.
• Setup:
  • V = \int_{0}^{2} \pi\big[(3 - x)^{2} - (1)^{2}\big]dx
• The integral evaluates to \frac{20\pi}{3} units³.

Common Pitfalls, Graphing & AP® Exam Tips

• Forgetting to square the radii; the error doubles when subtracting.
• Mixing dx and dy. Always match slice direction with differential.
• Dropping or swapping bounds after switching variables.
• Use a graphing calculator to shade the region and trace distances. That visual check often saves points.

Quick Reference Vocabulary Chart

Term	Meaning
Disc	Solid circular slice with no hole
Washer	Circular slice with a central hole
Radius	Distance from axis to curve
Outer Radius	Larger distance in washer setup
Inner Radius	Smaller distance in washer setup
Axis of Rotation	Line the region revolves around
Definite Integral	Adds infinitely many slices for exact volume
Bounds	Start and end values of integration
Cross Section	Shape cut perpendicular to axis

Conclusion

The disc method vs washer method boils down to one observation: does the region touch the axis? If it does, use discs; if not, washers fill the gap. Remember the main formulas:

V_{\text{disc}} = \int \pi R^{2}d(\text{variable}) V_{\text{washer}} = \int \pi\big(R_{\text{outer}}^{2} - R_{\text{inner}}^{2}\big)d(\text{variable})

Therefore, always sketch first, label radii, choose dx or dy wisely, and double-check work with a calculator. With steady practice, mastering the disc method vs washer method becomes second nature and boosts both confidence and scores on the AP® Calculus exam.

Sharpen Your Skills for AP® Calculus AB-BC

Are you preparing for the AP® Calculus exam? We’ve got you covered! Try our review articles designed to help you confidently tackle real-world math problems. You’ll find everything you need to succeed, from quick tips to detailed strategies. Start exploring now!

• 8.11 Volume with Washer Method: Revolving Around the x- or y-Axis
• 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled (BC)

Need help preparing for your AP® Calculus AB-BC exam?

Albert has hundreds of AP® Calculus AB-BC practice questions, free responses, and an AP® Calculus AB-BC practice test to try out.