I literally watched your previous video yesterday and in the morning I used the volume method you taught to derive the sphere volume and by sheer coincidence youve uploaded this video today 😂
These types of "rotate around the axis" volume calculations were probably my favorite part of Calculus II I remember when were learning geometry volumes in 4th grade and we got to the volume of the cone = (1/3)*pi*r^2*h and I asked, how do we know that it is 1/3 of a cylinder? The teacher gave a good answer of, maybe you fill it with sand and see that it takes three of them to fill the cylinder, which is empirically ok, but I was very happy when I got to Calc II and could derive it analytically. I finally got my question answered exactly.
Is it possible to derive the formula for the surface area of a sphere in the same way? Just instead of calculating the area of the entire slice of a semi-circle, we calculate the length of the hypotenuse of triangles which we then make progressively more narrow so they end up giving us the arclength of the semi-circle?
You're starting with the assumption that the formula for volume of a disk is known. Why not assume nothing and use polar coords? That's the way we learned it. And when you convert the Pythagorean formula to polar you get triple int of rho^2 * sin(phi) d(rho) d(theta) d(phi) with rho from 0 to r, theta from 0 to 2*pi, and phi from 0 to pi.
What is the curvature?? Is it measured in degrees or radians? Is it the radius of the circle that has its center on a line perpendicular to the tangent of the curve at the specified point?
I love the Tau vs Pi argument because it's fun to think about... but it was the comparison of the derivation of the volume of a sphere, to other well known equations that come to us through integration, that made me a believer in Tau.
Can this be done without calculus though, I wonder. When I asked my calculus teacher they said that this wasn’t the first/original way they came up with the formula, so I have been wondering how you could prove it without calculus.
Check out the volume of a cone proof 👉 ua-cam.com/video/drpxZ1aztWE/v-deo.html
can you do more calculus proofs please? I am a geometry student and this is very interesting.
I literally watched your previous video yesterday and in the morning I used the volume method you taught to derive the sphere volume and by sheer coincidence youve uploaded this video today 😂
These types of "rotate around the axis" volume calculations were probably my favorite part of Calculus II
I remember when were learning geometry volumes in 4th grade and we got to the volume of the cone = (1/3)*pi*r^2*h and I asked, how do we know that it is 1/3 of a cylinder? The teacher gave a good answer of, maybe you fill it with sand and see that it takes three of them to fill the cylinder, which is empirically ok, but I was very happy when I got to Calc II and could derive it analytically. I finally got my question answered exactly.
That was so satisfying.
I feel like I'm looking at a beautiful artwork without understand total of his beauty. But I want someday understand
This is beautifull
Great interpretation...Sir...Thank You very much...
Calculus now makes sense.
Now do it without calculus
That's a topic for the other channel
Ok Zeno
Uncalculus my calculus pls
@@SuryaBudimansyah oh I didnt see this is the calc channel
Euclid be like:
next time could you make video for solve sphere volume with jacobian matrix ?
Is it possible to derive the formula for the surface area of a sphere in the same way? Just instead of calculating the area of the entire slice of a semi-circle, we calculate the length of the hypotenuse of triangles which we then make progressively more narrow so they end up giving us the arclength of the semi-circle?
in em field theory class we do this in spherical coordinates and I think it's easier to understand and visualize but you need to know vector algebra 🤓
Nice.
Cool!
+C 😅
at r=0, volume = 0.
So, C = 0.
Why the surface area of the ball is 4*pi*r^2
dV/dr = 4pi r^2; so 4pi r^2 dr gives you the volume of a hollow sphere of area 4pi r^2 and thickness dr.
This should be taught in the elementary school :P
I think everyone understands the pi*r^3 part.
Its the 4/3 part that is very strange.
👋
Now we need the same with a circle and circumference.
Pleeeeasseeee find Volume of 4th dimension sphere and 4d volume of 4d sphere 😢🙏🙏🙏
I believe that volume is not the unit you mean, but the answer is the Integral from 0 to r of 2 * (4/3) * π * (√(r^2 - x^2))^3 dx.
@@happend thenk you
@@Nobodyman181 I hope I've helped ^^
You're starting with the assumption that the formula for volume of a disk is known. Why not assume nothing and use polar coords? That's the way we learned it. And when you convert the Pythagorean formula to polar you get triple int of rho^2 * sin(phi) d(rho) d(theta) d(phi) with rho from 0 to r, theta from 0 to 2*pi, and phi from 0 to pi.
That's the approach I'd take.
Share the link, please
ha just did this video: "Volume of a sphere with a triple integral"
bro, why this pop up in my notifications? im getting PTSD from university calculus D:
What is the curvature?? Is it measured in degrees or radians? Is it the radius of the circle that has its center on a line perpendicular to the tangent of the curve at the specified point?
I love the Tau vs Pi argument because it's fun to think about... but it was the comparison of the derivation of the volume of a sphere, to other well known equations that come to us through integration, that made me a believer in Tau.
Were you able to solve the integral @bprp calculus basics?
The sphere coordinates are here:
x = t cos(u) cos(v)
y = t cos(u) sin(v)
z = t sin(u)
-pi/2
Can this be done without calculus though, I wonder.
When I asked my calculus teacher they said that this wasn’t the first/original way they came up with the formula, so I have been wondering how you could prove it without calculus.
Calculus, of this nature, is a short hand for a Limit Sum as ∆x->0
Now do surface area