Volume of a Solid of Revolution y-Axis

This lesson explains how to use the disk method to find the volume of a solid of revolution when the region is rotated about the y-axis.

Summary

The video will show you why horizontal slices are required, how the radius must be written as a function of y, and how the limits of integration come from y-values instead of x-values. Several examples show how to rewrite equations, identify the correct radius, and set up the volume integral correctly. This skill is essential when the axis of rotation is vertical and prepares you for more advanced AP Calculus volume problems.

Key Takeaways

• When rotating a region about the y-axis, horizontal slices are used.

• The thickness of each slice is delta y.

• The radius must be written as a function of y.

• The disk method volume formula is V = π ∫(radius)² dy.

• Limits of integration must come from the y-values of the region.

• Functions given in terms of x often need to be solved for x before integrating.

• Squaring the radius ensures all volume contributions are positive.

• Final answers are expressed in cubic units.

Transcript

All right everyone, Mr. Antonucci here. We’re continuing with Section 8.3 and using the disk method to find volume when revolving a region about the y-axis.