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Math Help - Solids of Revolution: Washer or Shell Method?

  1. #1
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    Solids of Revolution: Washer or Shell Method?

    Hello everyone. I have a general question in regards to the washer and shell method. When and how do I know when to use either one? I know HOW to use them - I just need to know when to determine which one to use. For example:

    1) Let R be the region bounded by the lines y = 0, y = x, and y = 6-x.

    a) Find the volume generated when R is rotated around the line y = -3.

    b) Find the volume generated when R is rotated around the line x = 0.

    Again, I don't want the solutions/answers. I want to know which method to use for each one, and WHY we choose that method as opposed to the other one. I know that in theory, both methods should work, but there are instances where one is clearly easier than the other. That is the concept I am having some difficulty to grasp.
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    I generally use shells when I'm rotating around an axis, though you can use shells in the other cases too, you just need to translate everything so that you end up rotating about an axis...
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    Quote Originally Posted by Prove It View Post
    I generally use shells when I'm rotating around an axis, though you can use shells in the other cases too, you just need to translate everything so that you end up rotating about an axis...
    I don't get what you're saying. For solids of revolution, you're always rotating around an axis, aren't you? My issue with this is how do I know whether I should use the washer or shell method - how can I tell, based on the problem, which method is more efficient to use.
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    It has to do with whether a function would intersect certain portions of your washer or shell twice or only once.

    For a), you could use either washers or shells. However, I would probably go with shells, because then I only have one y integral. If I use washers, I have to split up the x integral into two domains: 0<x<3 and 3<x<6.

    For b), it's the opposite: if I use shells, I have to split up the x integral over the two domains 0<x<3 and 3<x<6, whereas if I use washers, I just have one y integral.

    Does that make sense?
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    Quote Originally Posted by FullBox View Post
    I don't get what you're saying. For solids of revolution, you're always rotating around an axis, aren't you? My issue with this is how do I know whether I should use the washer or shell method - how can I tell, based on the problem, which method is more efficient to use.
    Never mind, I didn't understand your question properly...
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    Quote Originally Posted by Ackbeet View Post
    It has to do with whether a function would intersect certain portions of your washer or shell twice or only once.
    I know I sound pretty thick asking this, but how do I tell if it intersects my washer or shell a certain number of times?

    For a), you could use either washers or shells. However, I would probably go with shells, because then I only have one y integral. If I use washers, I have to split up the x integral into two domains: 0<x<3 and 3<x<6.
    I think I get what you're saying here. So I only have to worry about (y+3) instead of adding 3 to both f(x) and g(x)?

    For b), it's the opposite: if I use shells, I have to split up the x integral over the two domains 0<x<3 and 3<x<6, whereas if I use washers, I just have one y integral.
    I THINK I get it but I'm not sure. I guess my main question here [forgive my apparent dullness] is how do I determine when I have to have two seperate integrals?
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