Converge or Diverge

**Yogi_Bear_79** · November 30th 2006, 05:47 PM

Does $\int_{0}^{2}\frac{x}{\sqrt{4-x^2}}dx$ converges or diverge? If it converges, evaluate the integral.

**ThePerfectHacker** · November 30th 2006, 06:20 PM

Originally Posted by Yogi_Bear_79

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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