Deriving the Formula for the Volume of a Sphere | Calculus Application

[@rakeshupadhyay9082]

Thanks for the basics

[@pikabroiboy5093]

You just taught me basic calculus and something I've been wondering about geometry, thank you

[@bloodywingrich8844]

So cool!

[@rebeccachauhan9003]

This derivation was very hard when I personally tried to do it. Excellent work

[@englishforfunandcompetitio248]

There ain't sny redundant repeating. All he said and did was necessary. This video is meant for students and not for experts. To teach a student in an effective way, all this is necessary. Keep on doing the same way dude. I found it effective and NOT boring at all. Warm regards, Prof: Ahmad.

[@rickrys2729]

Thanks - something I could follow with basic integral calculus.

[@learnmathbydoing]

Thanks🥰🥰🥰 That's great to hear!

[@KipIngram]

Uh... it comes from circumference = pi*diameter. It's nothing but the simplest possible calculus. Let r be the radius of an arbitrary circle and let c be the circumference. You can envision your sphere as a stack of rings, with changing radius. Consider a hemisphere and let angle a measure the angle between horizontal and the radius R of a sphere you're building. The radius of a circle reached at angle a is just R*cos(a) The width an arc strip for that circle is just R*da. So the area of that strip is A = 2*pi*R*cos(a)*R*da A = 2*pi*R^2*cos(a)*da Now integrate that from 0 to pi/2. The integral of cos() is sin(), so we have Hemisphere area = 2*pi*R^2*sin(a), evaluate (0 to pi/2) The result is 2*pi*R^2. Double that to get the whole sphere - the surface area of a sphere is 4*pi*R^2. That's step 1. Now do a similar process adding up the volume of spherical subshells of an entire sphere. We just need to integrate from r=0 to r=R, the final sphere radius: Sphere volume = integral of 4*pi*r^2 dr from 0 ro R. The result, OF COURSE is (4/3)*pi*R^3. So there you go. Once you accept that the ratio of a circle's circumference to its diameter is a constant regardless of the size of the circle, and name that ratio pi, the rest is just simple math. There is no mystery here. And accepting the circumference /diameter thing isn't hard either, because THEY'RE BOTH LENGTHS. They're going to scale in exactly the same way.

[@learnmathbydoing]

Thanks for sharing your detailed breakdown of the formula derivation. It's always interesting to see different ways of approaching a problem.

[@longbeachboy57]

When I saw this the first time, I kind of realized what 'integration' was for...

[@BerndLercher]

Come to the point. Redundant repeating is boring.