Derive the Volume of a Sphere using Calculus

[Shivam]

Homework Statement

Derive the volume of sphere using Calculus. i saw videos on this topic on youtube, but i want to do it by the method of integrating a circle at angle θ (Theta) . i am posting a photo where i explained every thing i did but i couldn't know what i am doing wrong.

Homework Equations

Integrating a circle of radius r at angle θ (Theta)

The Attempt at a Solution

https://drive.google.com/open?id=1Va32w9eCJE_2nH4VIPqG8bFHRzcKuvIF[/B]

 

[Orodruin]

i am posting a photo where i explained every thing i did but i couldn't know what i am doing wrong.

No you did not. You posted a link to your google drive, which other people do not have access to.

Also, you should make the effort of typing out your attempt if you want people to help you.

 

[LCKurtz]

Your error is using $ds$ for the thickness of your disk. It should be $dy=R\cos\theta d\theta$.

 

[Dick]

Homework Statement

Derive the volume of sphere using Calculus. i saw videos on this topic on youtube, but i want to do it by the method of integrating a circle at angle θ (Theta) . i am posting a photo where i explained every thing i did but i couldn't know what i am doing wrong.

Homework Equations

Integrating a circle of radius r at angle θ (Theta)

The Attempt at a Solution

https://drive.google.com/open?id=1Va32w9eCJE_2nH4VIPqG8bFHRzcKuvIF[/B]

I can see your attempted solution. The problem is with $ds=Rd\theta$. That is arc length along the surface of the sphere. It is not the same as the thickness of your circular section. Can you correct it?

 

[Shivam]

Homework Statement

Derive the volume of sphere using Calculus. i saw videos on this topic on youtube, but i want to do it by the method of integrating a circle at angle θ (Theta) . i am posting a photo where i explained every thing i did but i couldn't know what i am doing wrong.

Homework Equations

Integrating a circle of radius r at angle θ (Theta)

The Attempt at a Solution

https://drive.google.com/open?id=1Va32w9eCJE_2nH4VIPqG8bFHRzcKuvIF[/B]

Your error is using $ds$ for the thickness of your disk. It should be $dy=R\cos\theta d\theta$.

I got the correct answer by using the correcct thickness you gave me , but i still don't know how did you get that, can you explain please.

 

[Dick]

I got the correct answer by using the correcct thickness you gave me , but i still don't know how did you get that, can you explain please.

The thickness is the vertical thickness of the slice. The arc length you have is not vertical, it's tangent to the sphere. So it makes a varying angle with the vertical as you move up the sphere. Use trig to turn that into a vertical distance.

 

[Shivam]

Your error is using $ds$ for the thickness of your disk. It should be $dy=R\cos\theta d\theta$.

Can you show me how did you get that, i thought all day but i can't get it.

 

[haushofer]

Can you show me how did you get that, i thought all day but i can't get it.

Well, you have $y=R \sin(\theta)$, so $\frac{dy}{d\theta}=R \cos(\theta)$, so rearranging the differentials gives $dy=R \cos(\theta)d\theta$. Also, try to understand this result from a geometrical point of view (i.e. how does a small increase in the angle influece the increase in y?)