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Math Help - Quick question about finite area / infinite area

  1. #1
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    Quick question about finite area / infinite area

    Here's a question I have...

    Let S be the region between the graphs y = \frac {1}{x} and  y  = \frac {1}{\sqrt {x}} between x = 0 and x = 1. Does S have finite area? Justify your answer.

    Now, so do I first do this...  \int_{0}^{1} \frac{1}{x} and I figured out that this ends up diverging to  \infty. So does this mean that S has an infinite amount of area? Please tell me if I'm doing this correctly. Thanks.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by larson View Post
    Here's a question I have...

    Let S be the region between the graphs y = \frac {1}{x} and  y  = \frac {1}{\sqrt {x}} between x = 0 and x = 1. Does S have finite area? Justify your answer.

    Now, so do I first do this...  \int_{0}^{1} \frac{1}{x} and I figured out that this ends up diverging to  \infty. So does this mean that S has an infinite amount of area? Please tell me if I'm doing this correctly. Thanks.
    You are taking
    \int_0^1 \frac{1}{\sqrt{x}}~dx
    from this infinite area.

    Are you subtracting an infinte area from an infinite area? This one's a toughy. What you are going to have to do is
    S = \int_0^1 \left ( \frac{1}{x} - \frac{1}{\sqrt{x}} \right ) ~dx

    S =  \lim_{a \to 0} \int_a^1 \left ( \frac{1}{x} - \frac{1}{\sqrt{x}} \right ) ~dx

    And express that as a limit that has the indeterminate form of \infty - \infty. Do you know how to find this limit? (You'll probably end up using L'Hopital's rule at some point to give you a pointer as to where to look up how to do this.)

    -Dan
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  3. #3
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by topsquark View Post
    You are taking
    \int_0^1 \frac{1}{\sqrt{x}}~dx
    from this infinite area.

    Are you subtracting an infinte area from an infinite area? This one's a toughy. What you are going to have to do is
    S = \int_0^1 \left ( \frac{1}{x} - \frac{1}{\sqrt{x}} \right ) ~dx

    S = \lim_{a \to 0} \int_a^1 \left ( \frac{1}{x} - \frac{1}{\sqrt{x}} \right ) ~dx

    And express that as a limit that has the indeterminate form of \infty - \infty. Do you know how to find this limit? (You'll probably end up using L'Hopital's rule at some point to give you a pointer as to where to look up how to do this.)

    -Dan
    I think I have turned the tide against L'hopitals! Hooray!
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Mathstud28 View Post
    I think I have turned the tide against L'hopitals! Hooray!
    No, I've always been a bad boy about that... (as I'm not nearly clever enough to not use it!)

    -Dan
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  5. #5
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by topsquark View Post
    No, I've always been a bad boy about that... (as I'm not nearly clever enough to not use it!)

    -Dan
    Hey......its not about mathematical aptitude...its about mathematical laziness
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