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Math Help - Surface area and Volume

  1. #1
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    Surface area and Volume

    A Circle C is given by the equation x^2 +(y-1)^2 = 1. It is rotated about the x-axis to give a surface of revolution called a torus.

    a) Find the total surface area of the torus.

    (b) Find the volume of the torus and verify that it is 2pi times the area of C.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    re: Surface area and Volume

    Have you covered the theorems of Guldin?
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  3. #3
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    re: Surface area and Volume

    No, what we have covered are that :

    the surface area is the integral of 2pi y (1+(dy/dx)^2)^1/2 dx, and the volume is the integral of pi y^2 dx. I have used these equations, integrated from -1 to 1 and got 2pi(2+pi) for the area, and 10pi/3 +pi^2. But I think I'm doing something wrong. Can you help me?
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    re: Surface area and Volume

    Better work in parametric equations C\equiv x=\cos t,\;y=1+\sin t\;(t\in[0,2\pi]) . Then, the area is A=2\pi\int_0^{2\pi} y\sqrt{(dx/dt)^2+(dy/dt)^2}\;dt=\ldots . Let's see what do you obtain.
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  5. #5
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    re: Surface area and Volume

    Quote Originally Posted by FernandoRevilla View Post
    Better work in parametric equations C\equiv x=\cos t,\;y=1+\sin t\;(t\in[0,2\pi]) . Then, the area is A=2\pi\int_0^{2\pi} y\sqrt{(dx/dt)^2+(dy/dt)^2}\;dt=\ldots . Let's see what do you obtain.

    When I integrated and used the limits 0 and 2pi, I got an answer of 4pi^2, but when I changed to 0 and pi, I got 2pi(pi+2) which is the same answer as when I worked it out using the cartesian equation rather than the parametric. Which one is correct?

    And what about the volume? I worked it out using parametric as well but got an answer of 2pi^2. The question says that it should be 2pi times the answer of the area...

    Can you help please?

    Thanks
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    re: Surface area and Volume

    Quote Originally Posted by perla View Post
    When I integrated and used the limits 0 and 2pi, I got an answer of 4pi^2
    Right.

    And what about the volume? I worked it out using parametric as well but got an answer of 2pi^2. The question says that it should be 2pi times the answer of the area...
    Right, the volume is V=2\pi^2=2\pi\cdot \pi=2\pi \cdot \textrm{Area}(C)
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