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Math Help - area and length

  1. #1
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    area and length

    How are these two equations equal: (one interpreted as area and the other length)

    2integral{-1 to 1} sqrt[1-x^2]dx and integral{-1 to 1} 1/sqrt[1-x^2]dx

    First I understand that the second equation if a circle of radius 1 centered at the origin...and the first equation I'm considering the semicircle int he half plane...but why would they be equal? If I remove the 2 infront of the first integral, would it be the area of the other's region? confused!
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  2. #2
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    Quote Originally Posted by latavee View Post
    How are these two equations equal: (one interpreted as area and the other length)

    2integral{-1 to 1} sqrt[1-x^2]dx and integral{-1 to 1} 1/sqrt[1-x^2]dx

    First I understand that the second equation if a circle of radius 1 centered at the origin...and the first equation I'm considering the semicircle int he half plane...but why would they be equal? If I remove the 2 infront of the first integral, would it be the area of the other's region? confused!
    2\int_{-1}^1 \sqrt{1-x^2} \, dx = \pi ... (unit circle area)

    \int_{-1}^1 \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin(1) - \arcsin(-1) = \frac{\pi}{2} - \left(-\frac{\pi}{2}\right) = \pi

    can't see how this is an arc length.


    just looks like two definite integrals that yield the same numerical values, that's all.
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  3. #3
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    Ahh! I see, Thank you!
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