Disk Method: Revolve About y-axis

This lesson explains how to use the disc method to find the volume of a solid formed by revolving a region around the y‑axis. The key focus is rewriting functions in terms of y, using horizontal strips, and setting up integrals with respect to y.

Summary

This video shows how and why the setup changes when rotating about the y‑axis rather than the x‑axis. The lesson walks through two complete examples: rotating the region bounded by the curve y = x³ and rotating the region bounded by y = ln(x), showing how to convert each function into a form appropriate for integration with respect to y. You will learn how to rewrite each function as x = g(y), set up the integral with respect to y, determine correct y‑bounds, and compute the resulting volume. This method is useful whenever vertical slices would create complicated or invalid radii, and horizontal slices simplify the geometry.

Key Takeaways

• When rotating around the y‑axis using the disk method, use horizontal strips instead of vertical strips.

• The function must be rewritten in terms of y so the radius is expressed as x = g(y).

• The thickness of each horizontal slice is dy.

• The volume formula becomes V = π ∫ from c to d of (radius(y))² dy.

• Limits of integration must match the y‑values that bound the region.

• Always evaluate the definite integral carefully to get the final volume.