Surface Area of a Sphere (equation derived with calculus)

The equation for the area of a sphere is derived by summing up small ring elements of area along its perimeter.

[haydenhattenbach]

Calculus is so satisfying because everything that you were ever told just “to remember” without the reason now has an explanation

[hellboy_____]

The way you hold your pen is just amazing

[MusicMidas]

None of my calculus instructors explained why a particular method works (disk, shell, Green's theorem, etc.). Instead they threw formulas at us that we were expected to memorize. As a consequence, I effectively learned only to follow directions and not to truly understand calculus. I appreciate that you have broken this down into simpler steps to show WHY and HOW the actual calculus is applied in this derivation. Thanks!

[nilaksh]

Forget everything, did anyone noticed the perfect circle ?

[CHRISTIANADOLFOPEREZRAMIREZ]

Thank you man, you are the best. I didn´t understand my professor and you made it crystal clear, you have a gift for teaching.

[georgegiani]

Very well done. From this problem solving exercise I can see he is a very good math teacher. Clear, concise lesson!

[martinzone]

The step aside at the end was a nice touch.

[ceoOO]

I agree... very clear... the confusion is probably in the circumference element ds and understanding where it came about... its analogous to the surface area of the round part of a cylinder..2(pi)rh.. but instead of h, we use the differential for the smallest change in circumference. Other confusion may come from the need to express the integral in terms of theta instead of one of the co-ordinate variables, x, y, z, etc.. knowing all the basic formulas for circles will make things much easier..

[TSthegamer]

Your way of making Integral symbol is very unique 😊

[rodrigo-vlbi]

fantastic, this is exactly the kind of video i was looking for, a purely mathematical explanation

[ptyptypty]

Please, Step aside so I can see how Much Trouble I really am IN!! lol.. just kidding, that was a GREAT efficient Explanation!! you are very GOOD... I feel the same about your Video on the Area of a Circle!!!.... and to think.. I"m only NOW discovering your videos.. FIVE years after you posted them!!.. thank you!!

[Kram_Nebula]

ad for some blood sucking website feeding off the pain of a younger generation mocking us for being poor, " Still watching math tutorial videos from 2006? *scoff* ". 2011 actually and still incredibly helpful, saved me $40 and didn't have to tear me down to teach me. People like the man in this video are what our legacy as a whole is built on, not power hungry fear mongers clamoring for status. Sometime soon a child living in a fly over state in Nowhere America, that doesn't have access to advanced STEM programs or facilities will be watching this very same video and through hard work and perseverance will bring something revolutionary into this world that makes us all better. Be the change you want to see and leave each day hoping today was your day to contribute to someone else's future.

[aparnabiswas]

You explanations are absolutely fantastic. Got a great help. Thanks

[JoshLaurence_ETTO]

How is your board so perfectly white?!

[fuahuahuatime]

This is a nice approach. You could also just use a double integral and solve for the surface area of a sphere. 

[realityversusfiction]

From The Circle to the Sphere = Simple Empirical Arithmetic


Pi = 22/7 as an improper/vulgar fraction, which translates to being 3 whole units of 7, and 1/7th of 1 whole unit remaining

Or 3 diameter lengths consisting of 7 measurement units, with one measurement unit (1/7th of one diameter) remaining

PROOF

Finding The Length Of One Degree Of A Circle

1. Use a line measuring 120 centimeters as a Diameter Line

2. Multiply the 120 cm Diameter Line by 3

3. The Circle is 360 cm long

4. The Circle has 360 degrees

5. Each Degree is 1 cm long

Finding The Area Of A Circle

3 X r2 (Three times radius squared)

Given a Square measuring 120 x 120 cm; the Square will have an area of 14. 400 square centimeters.

1. Use one right angle of the 120 x 120 cm square as a Diameter Line
2. Divide the 120 cm Diameter Line by 2, to give a Radius of 60 cm
3. Square the 60 cm radius: 60 cm x 60 cm = 3, 600 square centimeters
4. Multiply the 3, 600 square centimeters by 3 =
=
10, 800 square centimeters

Therefore the Area of the Circle is 3/4 that of the 120 x 120 cm 14, 400 square centimeter square

CONCURRENCE

The Ancient Sumerian masters of geometry and mathematics defined this empirical reality, more than 2000 years before the Alexandrian Museum Plagiarizing Greeks.

Reference: “Estimating The Wealth” Encyclopedia Britannica.

A Babylonian cuneiform tablet written some 3, 000 years ago treats problems about dams, wells, water clocks, and excavations. It also has an exercise in circular enclosures with an implied value of π pi = 3. The contractor for King Solomon's swimming pool, who made a pond 10 cubits across and 30 cubits around (1 Kings 7:23) used the same value.

Sumerian Method

1. Diameter 120 cm's

2. Circumference 360 cm's

3. Circumference squared 129, 600 sq cm's

4. Divide 129, 600 by 12 =

10, 800 sq cm's to the area of the circle

However one question for me does remain begging, did they also manage to achieve this?

Twelve Steps From The Cube, To The Sphere

Calculating the surface area and volume of a 6 centimeter diameter sphere, obtained from a 6 centimeter cube.

1. Measure the (a) cubes height to obtain its Diameter Line, which in this case is 6 centimeter’.

2. Multiply 6 cm x 6 cm to obtain the square area of one face of the cube; and also add them together to obtain the length of perimeter to the square face = Length 24 cm, Square area 36 sq cm.

3. Multiply the square area, by the length of diameter line to obtain the cubic capacity = 216 cubic cm.

4. Divide the cubic capacity by 4, to obtain one quarter of the cubic capacity of the cube = 54 cubic cm.

5. Multiply the one quarter cubic capacity by 3. to obtain the cubic capacity of the Cylinder = 162 cubic cm.

6. Multiply the area of one face of the cube by 6, to obtain the cubes surface area = 216 square cm.

7. Divide the cubes surface area by 4, to obtain one quarter of the cubes surface area = 54 square cm.

8. Multiply the one quarter surface area of the cube by 3, to obtain the three quarter surface area of the Cylinder = 162 square cm.

CYLINDER TO SPHERE

9. Divide the Cylinders cubic capacity by 4, to obtain one quarter of the cubic capacity of the Cylinder = 40 & a half cubic cm.

10. Multiply the one quarter cubic capacity by 3, to obtain the three quarter cubic capacity of the Sphere = 121 & a half cubic cm, to the volume of the Sphere.

11. Divide the Cylinders surface are by 4, to obtain one quarter of the surface area of the Cylinder = 40 & a half square cm.

12. Multiply the one quarter surface area by 3 to obtain the three quarter surface area of the Sphere = 121 & a half square cm, to the surface area of the Sphere

Confirmation by Weight

Given that the 6 Centimeter Diameter Line Sphere was obtained from a Wooden Cube weighing 160 grams, prior to it being turned on a wood lathe into the shape of a sphere

The Cylinder of the Cube would weigh 120 grams
The waste wood shavings would weigh 40 grams

Given that the Cylinder weighed 120 grams
The waste wood shavings would weigh 30 grams.

Note: And ironically you can also obtain this same result by volume, using Archimedes Principle.

[asriel]

For anyone wondering why s = R*Theta, it is the formula for the Arch length. Totally forgor 💀and got confused when decided to refresh some of calculus, and I am willing to bet someone is in the same predicament as I.

[LGFZ]

i got it clear but the way you explained was more clear and good

[swarnendupachal]

that drawing ❤️❤️,
Sir thank you very much ❤️❤️

[andrzejpawowski]

Why S is equal to R theta and what is S anyway?