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Math Help - Finding the Volume of a Torus

  1. #1
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    Finding the Volume of a Torus

    I am supposed to find the volume of a torus with radii r and R using cylindrical shells. I don't even know what a torus is, and the whole shell method is still confusing to me.
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  2. #2
    Super Member
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    The plot is a cross-section of a torus showing a cross-section of one of the cylindrical shells (thick black walls) we'll use to calculate the volume. The equation of the cross-section on the right is:

    (x-R)^2+y^2=r^2 or y=\sqrt{r^2+(x-R)^2}

    and the formula for calculating the volume using cylindrical shells is:

    V=2\pi\int_a^b x f(x)dx

    Now, in this case, that would be:

    V=2\left(2\pi\int_{R-r}^{R+r} x\sqrt{r^2-(x-R)^2}dx\right)

    since we're integrating only the top half right?

    which is messy and also you get an \arctan component which you have to take the limit of as x goes to either limit. Eventually you get:

    V=4\pi\left(1/4\pi r^2 R-(-1/4\pi r^2 R)\right)=2\pi^2 Rr^2

    Maybe an easier way though.
    Attached Thumbnails Attached Thumbnails Finding the Volume of a Torus-crosssectiontorus.jpg  
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