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Please help to calculate: $$\int\sqrt {{r}^{2}-{x}^{2}}{dx},\quad x\in[0,r]$$ Do any method of trigonometric substitution?

Thanks.

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2 Answers

up vote 5 down vote accepted

Use the substitution $x=r\sin t$.

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Good!. Thanks so much! –  mathsalomon Aug 25 '11 at 23:37
    
@mathsalomon: You are welcome! –  Américo Tavares Aug 25 '11 at 23:41
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For future reference. That difference of squares may remind you of a certain right triangle. Use it to find a good substitution to use. –  GEdgar Aug 26 '11 at 0:26
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Or use the substitution x = r cosh t if you like calculus on the complex plane.

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Hadamard said, "Le plus court chemin entre deux vérités dans le domaine réel passe par le domaine complexe." This is an example where that quote is not apropos. –  Jonas Meyer Aug 26 '11 at 0:42
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