Volume with Cross Sections
Proving Formula for Cone

This video shows you how to use calculus to prove the volume formula for a right circular cone using cross-sections and integration.

Summary

In this video you'll see that by positioning the cone conveniently and analyzing a typical slice perpendicular to the x-axis, the radius of each circular cross-section is written as a function of x using similar triangles. This allows the area of each slice to be expressed algebraically and integrated over the height of the cone. The process shows why the cone volume formula V = (1/3)πr²h is true, rather than something to memorize, and reinforces connections between geometry and integral calculus.

Key Takeaways

• The volume formula for a right circular cone can be derived using calculus.

• The cone is positioned with its vertex at the origin and its axis along the x-axis.

• Cross-sections perpendicular to the x-axis are circles.

• Similar triangles relate the radius of a cross-section to its distance x from the vertex.

Transcript

Introduction

Okay, Mr. Antonucci here, and in this video we are going to prove the formula for the volume of a right circular cone using calculus.