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Math Help - Converge or Diverge

  1. #1
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    Converge or Diverge

    Does \int_{0}^{2}\frac{x}{\sqrt{4-x^2}}dx converges or diverge? If it converges, evaluate the integral.
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  2. #2
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    Quote Originally Posted by Yogi_Bear_79 View Post
    Does \int_{0}^{2}\frac{x}{\sqrt{4-x^2}}dx converges or diverge? If it converges, evaluate the integral.
    This is an Improper integral of the Second Type.
    There is a vertical asymptote at x=2
    Thus, you need to find,
    \lim_{t\to 2^-}\int_0^t \frac{x}{\sqrt{4-x^2}} dx
    Use inner functional substitution,
    u=4-x^2
    Thus,
    \lim_{t\to 2^-} -(4-x^2)^{1/2}|_0^t=-(4-t^2)^{1/2}+(4-0^2)^{1/2}=2
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