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Math Help - Volume of a solid

  1. #1
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    Volume of a solid

    Hi, how do I solve this question?

    Let R be the region that is bounded by the triangle with the vertices (0,0), (3, 0), and (1, 1).

    Find the volume of the solid generated by revolving the region R about the line x = 3.

    Thanks.
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  2. #2
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    Quote Originally Posted by coldfire View Post
    Hi, how do I solve this question?

    Let R be the region that is bounded by the triangle with the vertices (0,0), (3, 0), and (1, 1).

    Find the volume of the solid generated by revolving the region R about the line x = 3.

    Thanks.
    sketch a diagram ...

    region is bounded by the lines y=x , y = 0 , and y = \frac{3-x}{2}

    rotating about the line x = 3, you can use cylindrical shells w/r to x (will require two integral expressions) or you can use disks w/r to y.
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  3. #3
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    If I use disks, would the integral be:

    \pi\int_{0}^{1}(3-2y)^2-y^2  dy

    ?
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  4. #4
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    Hi codfire
    Quote Originally Posted by coldfire View Post
    If I use disks, would the integral be:

    \pi\int_{0}^{1}(3-2y)^2-y^2  dy

    ?
    You need to consider the axis of rotation. It's rotated about x = 3, not x = 0.

    The formula : V=\pi \int^a_b (x-k)^2~dy , where k is the axis of rotation.

    And you also need to see which curves cover the larger volume after rotated, is it y=x or y = \frac{3-x}{2} ?
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