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Integrating to find the volume of sphere

  1. Apr 14, 2013 #1
    1. The problem statement, all variables and given/known data
    Hi I have a question about integrating to find the volume of a sphere but before that i need to tell all other results i got or it will not make sence.

    1. I needed to show how you get the formula for finding the volume of a sphere by the help of a cirkle with radius r . I used the rule "The volume of the solid formed by rotating the area between the curves of f(x) and g(x)"

    I solved it by integrating and got the result volume =4/3 π*r3

    2. I had to find the volume if the height h i from 0 to 2r. (which means the whole sphere)
    I know the result is (π/6)*h*(3*a2+h2), because the formular was in my book. But i need to use the same rule of rotating the area around the x axsis, but i dont know how i can do it that way?

    3. I had to find the radius if v is 2500. here i just put the v into the formular and isolated r which i got to be 8,4

    Besides that i need to show a graph of V as the function of h which i dont know how i can

    2. Relevant equations

    1. relevant formular v=(4/3)*π*r3
    2. relevant formularv=(π/6)*h*(3*a2+h2)

    3. The attempt at a solution
    i attemted to find the solution for nr.2 by integrating √r2-x2 since that is the normal ecuation for a circle.

    y=f(x)=∏∫ from 0 to 2r (√r2-x2 )^2 dx where i tried to make it shorter by trying to get rid of the ^2

    but when i did i got (√r2-x2 )-(√r2-x2 ) and to be honest i feel quite a bit lost in this

    for showing the graph i have to say that v(h) is on the y-axis and h is on the x axis but i dont get what is v(h) and how to find h alone in this

    please i would be gratefull for some help
     

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    Last edited: Apr 14, 2013
  2. jcsd
  3. Apr 14, 2013 #2

    BruceW

    User Avatar
    Homework Helper

    hmm. So you've already used rotation of the circle around the x-axis to get the full volume of the sphere. But now, the problem is that they want you to integrate only only up to a certain height, so the top of the sphere is going to get cut off, right? (I attached a picture of what I think it looks like, tell me if I've not got it right). So anyway, you need to find the area of this shape, then use rotation of that around the x-axis to get the volume of the object.

    So to begin with, you need to find the 2d area of a 'cut-off circle'. You don't need to think about volumes straight away. It still takes a bit of work to find the area of a 'cut-off circle' (If you haven't memorised the equation for it). And I should use the correct terminology, sorry. I think the bit being cut-off is called a circle segment, to use the proper word :) Anyway, I think there are a few ways to solve for it. Are you familiar with polar coordinates?
     

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