Lesson 10: 8.10 Disc Method: Revolving Around Other Axes
Duration of Days: 1
Lesson Objective
The student will be able to:
Construct a radius expression R(x) or R(y) when the axis of revolution is a line such as y=k or x=h.
Apply the "Top minus Bottom" or "Right minus Left" principle to determine the length of the radius relative to the shifted axis.
Determine whether the radius is based on the relative position of the curve and the line of revolution.
If we revolve a region around the line y=5 instead of y=0, how does the length of the radius change?
Does the order of subtraction inside the squared radius matter for the final numerical answer?
Why or why not?
How can you tell if an axis of revolution is "above," "below," "left," or "right" of the bounded region?
Axis of Revolution
Shifted Radius
Horizontal Axis of Revolution
Vertical Axis of Revolution
Boundary
Squared Radius
CHA-5.C: Calculate volumes of solids of revolution using definite integrals.
CHA-5.C.1: The volume of a solid of revolution around any horizontal or vertical axis can be found using the disc method.
Mathematical Practice 2.B: Identify mathematical information from graphical representations.
Description: This lesson teaches students that the radius R is simply a distance. When revolving around y=0, R is just f(x). When revolving around y=k, R becomes the absolute difference. Students must visualize the region and the axis to ensure they are measuring the distance correctly.
Purpose: The AP Exam rarely asks for a simple rotation around the x or y axis. By mastering "Other Axes," students gain the flexibility needed to handle any horizontal or vertical line. This also reinforces the "Top minus Bottom" logic used in area problems.
DOK Level 2 (Skill/Concept): Correcting the radius formula for a given line and computing the volume.
DOK Level 3 (Strategic Thinking): Visualizing the solid and its orientation when the axis of revolution is a boundary of the region versus when it is outside the region.
For Struggling Learners (Scaffolding):
The "Anchor" Visualization: Have students draw the axis of revolution as a thick dashed line. Then, draw a "radius stick" from that dashed line to the function.
Identify the Axis: Before starting the integral, have students write down "Axis: to clarify the two components of the radius.
For Advanced Learners (Extension):Parameterization: Give the axis of revolution as a variable k and ask students to find the value of k that would result in a specific volume.
Symmetry Check: Challenge students to revolve a region around y=k and y=-k to see if/when the volumes are identical.
AP College Board Assessments