1. ## Sphere Area

Hello,

Problem:
I can't find my mistake when solving the surface area of a sphere.

Description of the sphere:
Given a sphere of radius $\displaystyle R$ centered at the origin. There is a circle at distance $\displaystyle y$ from the origin which has a radius of $\displaystyle h = \sqrt{R^2 - y^2}$ which is inscribed in the sphere

Description of the problem:
I was solving both the volume and area of a sphere. At first I calculated the volume of the sphere in the following approach:
$\displaystyle S_{circle} = \pi h^2 = \pi (R^2 - y^2 )$ (Area of the circle)
$\displaystyle V = \int_{-R}^{R} \pi (R^2 - y^2 ) dy = \frac{4}{3} \pi R^3$
Where it calculates the area of every circle and multiplicates by the differential. No problems here.

Now I wanted to do the same for the surface area, but using the circumference length instead, therefore:
$\displaystyle C_{circle} = 2 \pi h = 2 \pi \sqrt{R^2 - y^2}$ (Length of the circumference)
$\displaystyle S = \int_{-R}^{R} 2 \pi \sqrt{R^2 - y^2} dy = \pi ^2 R^2 \neq 4 \pi R^2$

2. ## Re: Sphere Area

Originally Posted by Murchinfus
I can't find my mistake when solving the surface area of a sphere.
$\displaystyle \color{red}S = \int_{-R}^{R} 2 \pi \sqrt{R^2 - y^2} dy = \pi ^2 R^2 \neq 4 \pi R^2$
You have the wrong idea about surface area.
$\displaystyle S=2\pi\int_a^b {f\sqrt {1 + \left( {f'} \right)^2 } }$

3. ## Re: Sphere Area

Thanks for your answer. I'm sorry but I think I was not very clear.
My question is why is it that this logic for calculating the volume of the sphere is not also valid for calculating the area of the surface of the sphere.

4. ## Re: Sphere Area

Originally Posted by Murchinfus
Thanks for your answer. I'm sorry but I think I was not very clear. My question is why is it that this logic for calculating the volume of the sphere is not also valid for calculating the area of the surface of the sphere.
Because volume and area are two entirely different concepts.

As the formula I gave you shows surface area involves arc-length.

5. ## Re: Sphere Area

But is it not true that $\displaystyle 2 \pi \sqrt{R^2 - y^2} dy$ is a differential area on the surface of the sphere?
It is probably not, but I just can't figure geometrically why.

6. ## Re: Sphere Area

Originally Posted by Murchinfus
But is it not true that $\displaystyle 2 \pi \sqrt{R^2 - y^2} dy$ is a differential area on the surface of the sphere?
It is probably not, but I just can't figure geometrically why.

7. ## Re: Sphere Area

Originally Posted by Plato
Ok, I saw in that page a formula on how to calculate the arc length using integration. That could be a way to calculate the lenght of a circumference in the sphere, but still I don't see how that apply to this problem. I am not trying to apply that formula or the one you gave me, I would like to do it purely geometric.

And does it hold or not that $\displaystyle 2 \pi \sqrt{R^2 - y^2} dy$ is a differential area in the sphere?

8. ## Re: Sphere Area

Originally Posted by Murchinfus
Ok, I saw in that page a formula on how to calculate the arc length using integration. That could be a way to calculate the lenght of a circumference in the sphere, but still I don't see how that apply to this problem. I am not trying to apply that formula or the one you gave me, I would like to do it purely geometric. And does it hold or not that $\displaystyle 2 \pi \sqrt{R^2 - y^2} dy$ is a differential area in the sphere?
Who is the author of your calculus textbook?

9. ## Re: Sphere Area

I used James Stewart's book and another one for series and differential equations, which I don't record right now.
I found a video explaining the method I tried to use.

10. ## Re: Sphere Area

Originally Posted by Murchinfus
I used James Stewart's book and another one for series and differential equations, which I don't record right now.
I found a video explaining the method I tried to use.
But that is on volume not surface area.
You posted a question on surface area.
Did you mean volume all along?

11. ## Re: Sphere Area

No no, I meant I am using the same logic.
One can use use the sum of the areas of disks to calculate the full volume of a sphere,
So I would like to use the sum of the length of circumferences to calculate the area of the surface, but this does not work, and this is where I don't know what is wrong.

12. ## Re: Sphere Area

In my copy of Stewart, in the chapter on Further Applications of Integration in section 2, surface area is discussed.