Why Surface Area of Sphere is 4πR^2 | Prove Area of Circle is πR^2 | Important Concept of Calculus
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- čas přidán 8. 09. 2024
- Why Surface Area of a Sphere is 4πR^2 | How to Find Area of a Circle | Important Concept of Calculus
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/ @mathbooster
It's like, Your first proving video sir.. :)
I also know that volume of a sphere is (4/3)πr³. Sir, can you make a video proving that too?
Consider a sphere of radius r and thickness dr then it's volume will be integral of 4πr² dr i.e sumof volume of all spheres of dr thick ness.integrate from 0 to R you will get the answer
From now I know how the formula comes😊
Sir I have solved the second part of deriving area of sphere where the if one part is considered to be height then we can have the radius and height increasing at same rate and then integrate this leads to same answer when considered from top
How is this same rate ?
r/h=tanx(x is the angle between the radius of sphere and height]
dr/dh=tanx (dx/dh)+hsec²x [x is a function of h]
The fact is that height is increasing when the radius of strips are decreasing
@@KM-om1hm right one is increasing and another one is decreasing 😂😂
I am interested this long time 🥰
Nice Work, Keep it UP
Sir, but Rdtheta , from where it come
But where is (RdΘ ) come from?
It is length of arc
Can sir also prove the surface area 🥺🙏🏻❓
But the length of the top of the cut strip circumference is different at the top and bottom. It isn’t a rectangle at all.
dx means infinitesimally small change in x, so we can simply neglect the change dx while calculating the area of rectangle.
@@MathBooster ah. That was never stated. Question; if it is infinitesimally small, then it does not have an area within it, no?
Can we do that with partial derivatives?
I need to know what are they. I only know integrals, summations, trigonometry, derivatives, limits, algebra. What are partial derivatives?
@@OlgaPlaysMatch-3Game Partial derivative means taking the derivative of a function with respect to one variable while keeping all other variables constant . For example let's say you have a function z=f(x,y). The partial derivative with respect to x would be done by treating all y terms as constants and then we differentiate as usual
Like b²-ac>0
∂2u∂ξ∂η+...=0 ∂ 2 u ∂ ξ ∂ η + . . . = 0
This is wave propagation form of a partial differential equation
Sphere volume is 4/3(pi)r3 when you derive it by r you get the surface 4(pi)r2. Why? do you have a proof of this.
I don't understand. What you want to say ?
I mean When you derive the volume of sphere you get the surface why?
No, you will not get surface. You might be doing it wrong way. For surface we can take rings as elementary part, but for volume we need to take disc as elementary part. So the whole calculations will be different.
It is tottaly seperate question from above video....I mean whe we derive volume of sphere by r we get the surface . When we derive the surface of the circle by r we get the perimeter. I am just wondering if there is a connection.
@@ulusoyem It's a very interesting point. You can work it the opposite way. Think of the sphere as made up of concentric shells with radius x and area 4 pi x^2
The volume is the integral of this area times the radius element dx i.e.
Volume = (integral from x = 0 to r) 4 pi x^2 dx
= (4/3) pi r^3
Oh 😍
Are you physicist?
First of all, who asked?
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