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Math Help - Sphere Area

  1. #1
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    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 R centered at the origin. There is a circle at distance y from the origin which has a radius of  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:
    S_{circle} = \pi h^2 = \pi (R^2 - y^2 ) (Area of the circle)
    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:
    C_{circle} = 2 \pi h = 2 \pi \sqrt{R^2 - y^2} (Length of the circumference)
    S = \int_{-R}^{R} 2 \pi \sqrt{R^2 - y^2} dy = \pi ^2 R^2 \neq 4 \pi R^2
    Last edited by Murchinfus; November 7th 2011 at 08:16 AM.
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  2. #2
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    Re: Sphere Area

    Quote Originally Posted by Murchinfus View Post
    I can't find my mistake when solving the surface area of a sphere.
    \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.
    S=2\pi\int_a^b {f\sqrt {1 + \left( {f'} \right)^2 } }
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  3. #3
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    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.
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  4. #4
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    Re: Sphere Area

    Quote Originally Posted by Murchinfus View Post
    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.
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  5. #5
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    Re: Sphere Area

    But is it not true that 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.
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  6. #6
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    Re: Sphere Area

    Quote Originally Posted by Murchinfus View Post
    But is it not true that 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.
    Study this page.
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  7. #7
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    Re: Sphere Area

    Quote Originally Posted by Plato View Post
    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 2 \pi \sqrt{R^2 - y^2} dy is a differential area in the sphere?
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  8. #8
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    Re: Sphere Area

    Quote Originally Posted by Murchinfus View Post
    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 2 \pi \sqrt{R^2 - y^2} dy is a differential area in the sphere?
    Who is the author of your calculus textbook?
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  9. #9
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    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 explaining the method I tried to use.
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  10. #10
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    Re: Sphere Area

    Quote Originally Posted by Murchinfus View Post
    I used James Stewart's book and another one for series and differential equations, which I don't record right now.
    I found a 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?
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  11. #11
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    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.
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  12. #12
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    Re: Sphere Area

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