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.
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.
Muito bom, mestre! O senhor integra com base em coordenadas cilíndricas.
Thanks for the basics
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.
Thank you so much for your kind words, Professor Ahmad! 😊
Interesting video! (and English voice)
You just taught me basic calculus and something I've been wondering about geometry, thank you
🙏🏻👍🏻👍🏻👍🏻❤️❤️❤️
This derivation was very hard when I personally tried to do it. Excellent work
Thanks - something I could follow with basic integral calculus.
Thanks🥰🥰🥰 That's great to hear!
So cool!
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.
Thanks for sharing your detailed breakdown of the formula derivation. It's always interesting to see different ways of approaching a problem.
When I saw this the first time, I kind of realized what 'integration' was for...
भास्कराचार्य यांनी हीच पद्धत वापरून 'स्फिअर व्हॉल्यूम'' काढला,ईसवी सन १११०
Come to the point. Redundant repeating is boring.