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Math Help - integration/signed area question

  1. #1
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    integration/signed area question

    Find the area enclosed by the curve f(x)= 1-1/x^2, the x-axis, the line x=1/2 and x=2.

    For signed areas, do I take the absolute value of sections with negative areas or do I add all of the areas up as they are? For this particular question I get a final answer of A=0 by adding the areas as they are but the answer in the textbook is A=1 (and this is found by taking the absolute of the negative area between [1/2, 1] ) I'd like to be clarified on which method to use ...From what I understand, signed areas do not take absolute values of negative areas. What am I doing wrong?
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  2. #2
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    Quote Originally Posted by shawli View Post
    Find the area enclosed by the curve f(x)= 1-1/x^2, the x-axis, the line x=1/2 and x=2.

    For signed areas, do I take the absolute value of sections with negative areas or do I add all of the areas up as they are? For this particular question I get a final answer of A=0 by adding the areas as they are but the answer in the textbook is A=1 (and this is found by taking the absolute of the negative area between [1/2, 1] ) I'd like to be clarified on which method to use ...From what I understand, signed areas do not take absolute values of negative areas. What am I doing wrong?
    A = -\int_{\frac{1}{2}}^1 1 - \frac{1}{x^2} \, dx + \int_1^2 1 - \frac{1}{x^2} \, dx
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  3. #3
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    Find

     \int_1^2 1 - \frac{1}{x^2} dx+\left|\int_{\frac{1}{2}}^1 1 - \frac{1}{x^2} dx \right|
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  4. #4
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    Thank you.

    What about if the boundaries were between the lines y=-1/2 and y=1/2?
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