Volume with Cross Sections
Proving Formula for Sphere

This video show you how to use calculus to prove the volume formula for a sphere using cross-sections and integration.

Summary

In this video, you'll that by placing the sphere on a coordinate plane and slicing it perpendicular to the x-axis, each cross-section becomes a circle. Using the equation of a circle, the radius of each slice is written as a function of x. Integrating the area of these circular slices from −R to R (or using symmetry from 0 to R) produces the familiar formula V = 4/3 πR³. This proof connects geometry, algebra, and integral calculus in a single argument.

Key Takeaways

• The sphere is centered at the origin on the x–y coordinate plane.

• The radius of the sphere is R, extending from −R to R on both axes.

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

• The radius of each circular slice is given by the y-value at that x.

• The equation x² + y² = R² is used to write y in terms of x.

• The area of each cross-section is π(R² − x²).

Transcript

Introduction

All right guys, Mr. Antonucci here, and in this video we’re going to prove the formula for the volume of a sphere using calculus.

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