Volume of a Solid of Revolution

This lesson introduces the disk method for finding the volume of a solid of revolution with respect to the x-axis.

Summary

The video will show you how to model the volume of three‑dimensional solids formed by rotating a region around the x‑axis. The lesson explains how a function represents the radius of circular cross‑sections and how summing the volumes of thin disks leads to a definite integral. Several examples show how the disk method works even when a function is partially below the x‑axis and how to correctly identify the radius and limits of integration. This technique is foundational for AP Calculus applications involving volume.

Key Takeaways

• The disk method is used to find volumes of solids formed by rotating a region around an axis.

• A vertical slice rotated about the x‑axis forms a circular disk.

• The volume of each disk is π times the radius squared times the thickness.

• The radius is the distance from the axis of rotation to the graph.

• The volume formula is V = π ∫(radius)² dx.

• The disk method works even if the function is negative on part of the interval.

• Squaring the radius makes all contributions to volume positive.

• Units for volume are cubic units.

Transcript

All right everyone, Mr. Antonucci here. This is Section 8.3, volume of a solid of revolution. We’re going to talk about disks and the disk method.