Sphere volume illustrated #maths #geometry #geometrydash

Summary

The video derives the volume formula for a sphere by slicing it into thin circular disks and adding their volumes using an integral.

Derivation

Consider a sphere of radius (R), sliced perpendicular to an axis through its center. At position (x) along the axis, each slice is a circle with radius (r). The sphere’s radius, the slice radius, and the distance (x) form a right triangle:

[ r^2 + x^2 = R^2 ]

Therefore,

[ r^2 = R^2 - x^2 ]

The area of a slice at position (x) is:

[ \pi r^2 = \pi(R^2 - x^2) ]

Give each circular slice a tiny thickness (dx). Its approximate volume is its area multiplied by its thickness:

[ \pi(R^2 - x^2)\,dx ]

Adding all the slices from one end of the sphere to the other, from (x=-R) to (x=R), gives:

[ V = \int_{-R}^{R} \pi(R^2 - x^2)\,dx ]

Evaluating the integral:

[ V = \pi\left[R^2x - \frac{x^3}{3}\right]_{-R}^{R} = \frac{4}{3}\pi R^3 ]

A sphere with radius (R) has volume:

[ \boxed{\frac{4}{3}\pi R^3} ]

Speakers and sources

• Narrator: One speaker explains the derivation; no name is given in the subtitles.
• Video/channel source: Sohil Rathi.
• Website mentioned in the subtitles: “limmers.com” (the spelling may be uncertain because the subtitles are auto-generated).