Lesson 11: 8.11 Washer Method: Revolving Around the x- or y-Axis
Duration of Days: 2
Lesson Objective
The student will be able to:
Identify when a solid of revolution contains a hole, necessitating the Washer Method rather than the Disc Method.
Determine the Outer Radius (R) and the Inner Radius (r) by measuring the distance from the axis of revolution to the "far" curve and "near" curve.
Set up the volume integral using the formula.
Avoid the "Universal Squaring Error" by ensuring they square the radii individually rather than squaring the difference.
How does the presence of a gap between the region and the axis of revolution change the shape of a single "slice"?
Why must we subtract the "empty" volume from the "total" volume to find the volume of the solid?
Washer
Outer Radius (R)
Inner Radius (r)
Hole / Gap
Compound Solid
Annulus (Area of a Washer)
CHA-5.C: Calculate volumes of solids of revolution using definite integrals.
CHA-5.C.2: The volume of a solid of revolution with a hole can be found using the washer method.
Mathematical Practice 1.E: Apply appropriate mathematical rules or procedures.
Description: The Washer Method is essentially the Disc Method applied twice. To find the volume of a solid with a hole, we find the volume of the "outer" solid (using the curve furthest from the axis) and subtract the volume of the "inner" empty space (using the curve closest to the axis).
Purpose: This is one of the most common volume types on the AP Exam. It requires a higher level of spatial reasoning than the Disc Method, as students must correctly identify which function is "on top" relative to the axis of rotation to determine which is R and which is r.
DOK Level 2 (Skill/Concept): Identifying the two radii and evaluating the integral for a standard x or y axis rotation.
DOK Level 3 (Strategic Thinking): Determining the intersection points to find limits and correctly assigning R and r for a region where the functions might cross.
For Struggling Learners (Scaffolding):The "Two-Stick" Method: Have students draw two different colored lines from the axis of revolution. The long stick is R, and the short stick is r.
Formula Mnemonics: Use "Big R" and "Little r" to emphasize that the order of subtraction matters (Outer minus Inner).
For Advanced Learners (Extension):Volume of a Pipe: Have students use the Washer Method to derive the formula for the volume of a hollow cylinder (pipe) with length L, outer radius R, and inner radius r.
Revolving "Around the Region": Give them three curves that bound a region and ask them to find the volume when rotated around each of the three curves individually.
AP College Board Assessments