Why is the volume of a sphere V=4/3*pi*r^3? (calculus disk method)

This tutorial covers why the sphere with radius r has the volume V=4/3*pi*r^3. We will prove this by using the disk method for the volume of a solid of revolution. This is an application of integration that you will learn in your Calculus 2 class.

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[AncientBulldozer]

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 😂

[flowingafterglow]

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.

[IPi]

Thank you sir! I was not able to remember the formulas! Now i can just derive them!

[Sgth]

Now we need the same with a circle and circumference.

[МаксимАндреев-щб]

The sphere coordinates are here:
x = t cos(u) cos(v)
y = t cos(u) sin(v)
z = t sin(u)
-pi/2 <= u <= pi/2, 0 <= v <= 2pi, 0 <= t <= r.
The absolute value of Jacobian determinant is J = t^2 cos(u).
So the volume of the sphere is
the integral of t^2 cos(u) dt du dv = (R^3/3 - = R^3/3•2•2pi = 4/3 pi R^3.

[sanjidrohan]

What is the dv or how is the volume changing by very small?

[stefanmi]

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.

[ernestschoenmakers]

Is it coincidence that if you differentiate (4/3)*pi*r^3 wrt r you'll get 4*pi*r^2 the surface area of a sphere?

[professorsogol]

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?

[JakkAuburn]

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?

[Pappu-rdq]

Great interpretation...Sir...Thank You very much...

[SokomoKudiomi-hl]

I feel like I'm looking at a beautiful artwork without understand total of his beauty. But I want someday understand

[mehmetalivat]

next time could you make video for solve sphere volume with jacobian matrix ?

[sinekavi]

Were you able to solve the integral @bprp calculus basics?

[LuigiElettrico]

This should be taught in the elementary school :P

[megumn]

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 🤓

[BleuSquid]

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.

[hibosmo]

bro, why this pop up in my notifications? im getting PTSD from university calculus D:

[Ninja]

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.

[DRakont]

Pleeeeasseeee find Volume of 4th dimension sphere and 4d volume of 4d sphere 😢🙏🙏🙏