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Thread: Higher Level Integration Marathon & Questions

  1. #76
    Loquacious One Drsoccerball's Avatar
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    Re: Extracurricular Integration Marathon

    How do double integrals even work ?

  2. #77
    Ancient Orator leehuan's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Drsoccerball View Post
    How do double integrals even work ?
    You just evaluate them variable by variable.

    How you generate them is a more interesting story.
    Paradoxica likes this.

  3. #78
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Drsoccerball View Post
    How do double integrals even work ?
    Quote Originally Posted by leehuan View Post
    You just evaluate them variable by variable.

    How you generate them is a more interesting story.
    Yeah, knowing when to revert a function into the definite integral of a simpler function is not at all obvious for the most part.
    Last edited by Paradoxica; 21 Feb 2016 at 2:54 PM.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  4. #79
    Retired Carrotsticks's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Drsoccerball View Post
    How do double integrals even work ?
    You're probably clever enough to get the general gist of it (double integration via Cartesian coordinate system) from this image. I'm guessing you just want a rough idea, not a more rigorous construction.

    Just imagine throwing in an extra dimension to everything. Instead of 1 dimensional partitions (the subintervals) of a domain, we have 2 dimensional partitions of a region. Instead of taking 2 dimensional 'strips', we have 3 dimensional 'strips'. Instead of swooping over the domain once, we swoop over it twice (once in the x direction, again in the y direction). Instead of finding an 'area', we find a 'volume'.

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  5. #80
    Loquacious One Drsoccerball's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Carrotsticks View Post
    You're probably clever enough to get the general gist of it (double integration via Cartesian coordinate system) from this image. I'm guessing you just want a rough idea, not a more rigorous construction.

    Just imagine throwing in an extra dimension to everything. Instead of 1 dimensional partitions (the subintervals) of a domain, we have 2 dimensional partitions of a region. Instead of taking 2 dimensional 'strips', we have 3 dimensional 'strips'. Instead of swooping over the domain once, we swoop over it twice (once in the x direction, again in the y direction). Instead of finding an 'area', we find a 'volume'.

    Is this apart of Matrices ? Doesn't seem to hard :P So can we treat one of the changing functions as a constant while we integrate the 'flat' part ?

  6. #81
    Retired Carrotsticks's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Drsoccerball View Post
    Is this apart of Matrices ? Doesn't seem to hard :P So can we treat one of the changing functions as a constant while we integrate the 'flat' part ?
    Typically it's taught in an first course in vector calculus, not linear algebra where matrices tend to live.

    The initial treatments tend to be highly elementary in nature and are very much as you have described above. A Year 11 student could compute some double integrals, simply treating them as a "2 questions in 1" style problem.

    However, the difficulty usually comes in the construction of the integral and then spotting clever substitutions/manipulations to invoke Fubini or something that will help simplify the computation.

  7. #82
    Loquacious One Drsoccerball's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Carrotsticks View Post
    Typically it's taught in an first course in vector calculus, not linear algebra where matrices tend to live.

    The initial treatments tend to be highly elementary in nature and are very much as you have described above. A Year 11 student could compute some double integrals, simply treating them as a "2 questions in 1" style problem.

    However, the difficulty usually comes in the construction of the integral and then spotting clever substitutions/manipulations to invoke Fubini or something that will help simplify the computation.
    Like the first question on this marathon ? Thanks Carrot!

  8. #83
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Drsoccerball View Post
    Like the first question on this marathon ? Thanks Carrot!
    Pretty much. Though for that question we'd use a combination of the polar and cartesian systems to evaluate it.

    In the Volumes problem Q14 of 2014 BOS Trials Extension 2, I provided a VERY brief glimpse into evaluating that integral via double integration (Cartesian).

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    Ancient Orator leehuan's Avatar
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    Re: Extracurricular Integration Marathon

    I think I ignored all the volumes questions in the BoS trials...they looked scarier than inequalities and mechanics...

  10. #85
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Carrotsticks View Post
    Pretty much. Though for that question we'd use a combination of the polar and cartesian systems to evaluate it.

    In the Volumes problem Q14 of 2014 BOS Trials Extension 2, I provided a VERY brief glimpse into evaluating that integral via double integration (Cartesian).
    Yes, I didn't even pick that up when Sy123 did it for us :P

  11. #86
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  12. #87
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon



    Last edited by Paradoxica; 14 Mar 2016 at 10:03 PM.

  13. #88
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Prove the following result:

    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  14. #89
    Ancient Orator leehuan's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Paradoxica View Post
    Prove the following result:

    Was this the last question?

  15. #90
    Ancient Orator leehuan's Avatar
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    Re: Extracurricular Integration Marathon

    Also, this was a perplexing question for a Q2...


  16. #91
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by leehuan View Post
    Was this the last question?
    No, go to like first or second page for q20
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  17. #92
    Supreme Member seanieg89's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by leehuan View Post
    Also, this was a perplexing question for a Q2...

    Swap the order of integration to arrive at

    This is killed by the substitution , and the result is
    leehuan and Paradoxica like this.

  18. #93
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by seanieg89 View Post
    Swap the order of integration to arrive at

    This is killed by the substitution , and the result is
    I finally get how swapping order works... it only took me four months...
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  19. #94
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Paradoxica View Post
    Prove the following result:

    Also, looking for real methods of proving the above. I have seen the complex method, but that's not helpful since I don't know anything about complex analysis yet.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  20. #95
    Senior Member integral95's Avatar
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    Re: Extracurricular Integration Marathon

    “Smart people learn from their mistakes. But the real sharp ones learn from the mistakes of others.”
    ― Brandon Mull

  21. #96
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by integral95 View Post


    Nothing Extracurricular about this. Just tedious.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  22. #97
    Ancient Orator leehuan's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Paradoxica View Post


    Nothing Extracurricular about this. Just tedious.
    You forget that when they give it as cosh the HSC students don't know what it is.

  23. #98
    Senior Member integral95's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Paradoxica View Post


    Nothing Extracurricular about this. Just tedious.
    wow I can't believe I didn't take out a v when I did it,
    “Smart people learn from their mistakes. But the real sharp ones learn from the mistakes of others.”
    ― Brandon Mull

  24. #99
    -insert title here- Paradoxica's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by leehuan View Post
    You forget that when they give it as cosh the HSC students don't know what it is.
    That's just information. The difficulty of the integral is completely within the realms of the HSC.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  25. #100
    Ancient Orator leehuan's Avatar
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    Re: Extracurricular Integration Marathon

    Quote Originally Posted by Paradoxica View Post
    That's just information. The difficulty of the integral is completely within the realms of the HSC.
    You'd have to define cosh(x) if this were to be put in the MX2 integration marathon.

    Of course, if this was done in advance then I'd agree with what you mean by difficulty.

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