Volume of a Sphere (equation derived with calculus)

[@TheIndieGhost]

Another way to do it is to use slices of the sphere itself. *Setup* Let a sphere with radius R origin lay at the origin point of a three-dimensional grid. This means that along any axis, the sphere will go from -R to R. *Process* We can divide the sphere into infinitely thin slices, such that each slice is a circle. We can define a circle on the x axis having an area of πr^2, where r = √(R^2 - x^2) where x is in the range [-R, R]. We can redefine the circle as having an area of π(R^2 - x^2). *Integration* A(x) = π(R^2 - x^2) V = ∫π(R^2 - x^2) dx from x = -R to x = R. Note that R^2 is a constant, as R is the radius of the sphere. Because we are working with a sphere, the integral from -R to 0 will equal the integral from 0 to R. We can rewrite this as: V = 2π∫(R^2 - x^2) from x = 0 to x = R. Integrating, we get: V = 2π[R^2 * x - x^3 / 3] from x = 0 to x = R. V = 2π(R^3 - R^3 / 3) V = 2π(2R^3 / 3) V = 4πR^3 / 3

[@gamer_asurn1448]

Woe Thanks❤️❤️❤️

[@yi-chenhu7794]

Your videos are extremely helpful to math learner and the way you approach a problem is amazing! Conceptually simple and computationally easy. Highly recommended Mr. Remi's tutoring!

[@saulremi314]

胡屹鎮 thanks!

[@CHRISTIANADOLFOPEREZRAMIREZ]

Thank you so much, amazing videos. I hope you can do a playlist for calculus 3 or multivariable calculus videos. Greetings from Mexico.

[@jimitjain6748]

Please keep up the good work.... It helped me a lot in maths.. Keep making new videos further.

[@saulremi314]

Thanks

[@AkashRaj-pk4kx]

Why can't we use the same method for volume which you used in previous video of surface area of sphere

[@arthurlau2020]

Hi, I just watched your videos about calculating the surface areas of sphere. can I apply the same method on calculating the volume of sphere.

[@saulremi314]

I've never done it but you can try it and see. You'll need to sum up the area of circles. Like I said, I've never done it but it might work.

[@arthurlau2020]

@@saulremi314 I tried, but the answer was incorrect.

[@saulremi314]

@@arthurlau2020 I'm traveling right now but I'll give it a try when I get home in a few weeks.

[@saulremi314]

@@arthurlau2020 btw... I just changed the name of the channel. From "Best Damn Tutoring" to "Alpha Physicist". What do you think?

[@arthurlau2020]

@@saulremi314 sounds good. your new channel name is much more clear to the visitors.

[@JohnLee-xk4ox]

From n dimension to n+1, circle' circumstance to area, integral 2πrdr is πr^2. Great 👍

[@KM-om1hm]

What subject do you teach?

[@freeenergy8972]

area of circle is pi.R^2

[@saulremi314]

yes for circle but for sphere it's different=4*pi*R^2=area of sphere

[@koustubhtiwari5736]

@@saulremi314 sir why we took pi r ^2 if we have to find volume of sphere

[@saulremi314]

@@koustubhtiwari5736 bc we are summing up shells with a volume 4*pi*r^2*dr to get the total volume.

[@koustubhtiwari5736]

@@saulremi314 thank you sir

[@carl6167]

Here is my problem : You already use your knowledge of the area of the sphere shell. For the demonstation to be complete, you should also derive that...

[@Quelklef]

The volume of the shell is approximated by the surface area (4pir^2) times the differential (dr). The equation for the surface area we can accept as "already known", and the correctness of this approximation (especially in the context) is a concept of calculus itself. This video is just an application of said concept, so I think it's OK to assume the viewer already understands it.

[@antoniovelazquez9869]

He demonstrates it in a previous video: kzbin.info/www/bejne/gGWVqIGteM1jg80 this is a series of videos made by him about calculus.

[@qualquan]

Good if If we begin with SA

[@ilafya]

It’s easy when we know the result the question is why dV=Sdr

[@saulremi314]

dV=A*dr

[@channaiahchakravarthi3352]

Wooooo