Calculating Areas by Integration: Simple Problems

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  • Опубліковано 30 жов 2024

КОМЕНТАРІ • 7

  • @CornerstonesOfMath
    @CornerstonesOfMath  7 місяців тому

    The video about the GENERALIZED version of this problem is now uploaded. Check here: ua-cam.com/video/KAYdQu0FJHc/v-deo.html

  • @Convergant
    @Convergant 7 місяців тому +2

    You can generalise this nicely by using a surface integral instead of a single integral, and by making the substitution u = x^(1/n), y = v^(1/n). Then the Jacobian is n²×u^(n-1)×v^(n-1), and you can integrate with 0 ≤ v ≤ 1 and 0 ≤ u ≤ 1-v.
    Either way, you get that the nth area is n×Beta(n, n+1).

    • @CornerstonesOfMath
      @CornerstonesOfMath  7 місяців тому

      Yeah, thanks. I initially didn't thought on covering the general version due to it being somewhat advanced, but seeing that this video is gaining popularity, perhaps I should consider it......

  • @baptistebermond2082
    @baptistebermond2082 7 місяців тому +3

    These can be generalized to Int_0^1(dx(1-x^(1/n))^n)=Gamma(1+n)^2/Gamma(1+2n)

    • @CornerstonesOfMath
      @CornerstonesOfMath  7 місяців тому +2

      A fascinating general expression! However, I don't know if I will cover this topic in near future, because it is bit too advanced compared to the general level of math covered in my channel. Still, thank you for your contribution!

    • @blub232324
      @blub232324 7 місяців тому +1

      which is also 1 divided by the nth central binomial coefficient :)

    • @CornerstonesOfMath
      @CornerstonesOfMath  7 місяців тому

      @@blub232324 If n is natural number, then yes!