Lesson 9: 8.9 Disc Method: Revolving Around the x- or y- Axis
Duration of Days: 1
Lesson Objective
The student will be able to:Visualize a 3D solid generated by revolving a 2D region around the x-axis or y-axis.
Identify the radius R(x) or R(y) as the distance from the axis of revolution to the curve.
Apply the Disc Method formula to calculate volume.
Determine when to use dx (vertical discs) versus dy(horizontal discs) based on the axis of revolution.
How is a solid of revolution different from the cross-section solids we studied in the previous lessons?
Why is the area of a single "slice" in this method always a circle?
What happens to our integral setup if we switch from revolving around the x-axis to the y-axis?
Solid of Revolution
Axis of Revolution
Disc / Disk
Radius of Revolution
Horizontal vs. Vertical Partitioning
Constant of Integrability
CHA-5.C: Calculate volumes of solids of revolution using definite integrals.
CHA-5.C.1: The volume of a solid of revolution around a horizontal or vertical axis can be found using the disc method.
Mathematical Practice 1.E: Apply appropriate mathematical rules or procedures.
Description: The Disc Method is used when the region being revolved is "flushed" against the axis of rotation (meaning there is no gap/hole). Each slice of the solid is a solid cylinder (a disc) with a volume, where the radius r is the function's height and the width w is the differential dx or dy.
Purpose: This is a cornerstone of AP Calculus geometry. It teaches students to transform 2D functions into 3D models. It also reinforces the importance of the variable of integration: revolving around a horizontal axis requires dx, while a vertical axis requires dy.
DOK Levels
DOK Level 2 (Skill/Concept): Identifying the radius and evaluating the integral for a standard rotation around the x or y axis.
DOK Level 3 (Strategic Thinking): Determining the limits of integration for a region bounded by a curve and an axis and setting up the integral without a provided graph.
For Struggling Learners (Scaffolding):The "Radius Stick": Have students draw a line segment from the axis of revolution to the outer edge of the function. Label this "R." This helps them visualize that R = f(x) - 0
Reminder: Encourage students to write the pi outside the integral symbol immediately so they don't forget it in their final calculation—a very common error.
3D Animation: Use digital tools (like GeoGebra) to show the 2D area actually "sweeping" through space to form the solid.
For Advanced Learners (Extension):Deriving the Formula: Have students explain how the volume of a cylinder leads directly to the integral formula.
AP College Board Assessment