Lesson 12: 8.12 Washer Method: Revolving Around Other Axes
Duration of Days: 2
Lesson Objective
The student will be able to:
Construct expressions for both the Outer Radius (R) and Inner Radius (r) when the axis of revolution is a line other than y=0 or x=0.
Determine the relative distance of two functions from a shifted axis to correctly assign R (further away) and r (closer).
Apply the formula to find the volume of a solid with a hole revolving around any horizontal or vertical line.
Validate the radius setup by checking if the axis of revolution is "above/below" or "right/left" of the shaded region.
If the axis of revolution is y=10 and our region is near the x-axis, which function is "further" from y=10?
How does the "Top minus Bottom" rule change when the axis of revolution is actually between two functions? (Trick question: This rarely happens in standard problems, but it forces students to think about the gap!)
Why is it helpful to draw a "representative radius" from the axis to each curve before writing the integral?
Shifted Axis
Outer Radius R(x) or R(y)
Inner Radius r(x) or r(y)
Distance Formula (Calculus Context)
Horizontal vs. Vertical Rotation
Limits of Integration
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 2.B: Identify mathematical information from graphical representations.
Mathematical Practice 1.E: Apply appropriate mathematical rules or procedures.
Description: This lesson is the culmination of all volume techniques. Students must find the volume of a "hollowed-out" solid revolved around a line like y=-2 or x=4. The key skill is defining R and r as: horizontal and vertical axis.
Purpose: This is a high-probability topic for the AP Calculus Free Response Questions. It tests a student's ability to synthesize geometry, algebra, and integration. It requires them to move beyond memorized formulas and actually understand the spatial relationship between curves and lines.
DOK Levels
DOK Level 2 (Skill/Concept): Correcting the R^2 - r^2 setup for a shifted axis.
DOK Level 3 (Strategic Thinking): Setting up a volume for a region that must be integrated with respect to y because the rotation is around a vertical line x=h.
For Struggling Learners (Scaffolding):The "Stick" Method (Extended): Draw a dashed line for the axis. Draw a long arrow from the axis to the far curve (R) and a short arrow from the axis to the near curve (r).
Standardized Setup: Use the "Outside/Inside" check. R is the distance to the outside boundary of the solid; r is the distance to the inside boundary (the hole).
Color-coded Equations: Write the axis in red and the functions in blue/green to keep the "Top minus Bottom" subtraction clear.
Axis Displacement: Give them a region and ask: "Where would we place the axis y=k so that the resulting volume is exactly 100\pi?"
Multiple Rotations: Compare the volumes created by revolving the same region around y=k (above the region) vs. y=-k (below the region).
AP College Board Assessments