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honours maths-integration?

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  • 02-06-2008 8:17pm
    #1
    Desirah
    Closed Accounts Posts: 10 ✭


    Would anyone be able to tell me how to prove the area of a cylinder by integration or where i might find it.Just heard on that coundown to 406 show that it might come up. Help appreciated, thanks.


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    #2
    Cokehead Mother
    Closed Accounts Posts: 773 ✭✭✭Cokehead Mother


    Well you've got a line with two points, (0,r) and (h,r).

    So y = r.

    y^2 = r^2

    Integrate that from 0 to h and multiply it by pi.


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    #3
    Fuascailt
    Closed Accounts Posts: 268 ✭✭Fuascailt


    I heard that too, i was just thinking about it. If you rotated a line around the x-axis. But it would have to be y=k and you couldnt get formula out of that. You could do it for specifics. I have the syllabus somewhere, i must check if its on it.


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    #4
    Cokehead Mother
    Closed Accounts Posts: 773 ✭✭✭Cokehead Mother


    Fuascailt wrote: »
    I heard that too, i was just thinking about it. If you rotated a line around the x-axis. But it would have to be y=k and you couldnt get formula out of that. You could do it for specifics. I have the syllabus somewhere, i must check if its on it.

    It's on the syllabus. They can ask us to rotate a line or a circle.


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    #5
    Fuascailt
    Closed Accounts Posts: 268 ✭✭Fuascailt


    Ok, ignore me, her post came up while i was writing:D


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    #6
    Fuascailt
    Closed Accounts Posts: 268 ✭✭Fuascailt


    Thanks.


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    #7
    turgon
    Closed Accounts Posts: 3,762 ✭✭✭turgon


    Can they ask you to get the area of a circle be integration?


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    #8
    Cokehead Mother
    Closed Accounts Posts: 773 ✭✭✭Cokehead Mother


    turgon wrote: »
    Can they ask you to get the area of a circle be integration?

    Circular integrals are on the course (1999 Q8 c is an example of them) but they're never gonna ask say "Show that the area of a circle is pi.r^2 using integration methods".


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    #9
    Parsley
    Closed Accounts Posts: 3,144 ✭✭✭Parsley


    Cokehead Mother wrote: »
    Circular integrals are on the course (1999 Q8 c is an example of them) but they're never gonna ask say "Show that the area of a circle is pi.r^2 using integration methods".

    It'll most likely be volume of a sphere, cone or cylinder.


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    #10
    Cokehead Mother
    Closed Accounts Posts: 773 ✭✭✭Cokehead Mother


    It could technically be frustum. :eek:

    I don't think they'd be that mean though.


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    #11
    Parsley
    Closed Accounts Posts: 3,144 ✭✭✭Parsley


    They'd have to be an awful bunch of cúnts to pull that one out! What would ya do for it? Treat it like a cone but use limits of r and R instead or r and 0?


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    #12
    Cokehead Mother
    Closed Accounts Posts: 773 ✭✭✭Cokehead Mother


    Well you'd just get the equation of the line joining (0,r) and (h, R) and then get y^2 and integrate from 0 to h. It's straightforward but is the most tedious thing ever. The cone was a 10 marker last time it came up so I guess this could fit within the realms of a 20 marker LC question. Maybe the person who set the paper is a cúnt! :eek:


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    #13
    Parsley
    Closed Accounts Posts: 3,144 ✭✭✭Parsley


    Well if it comes up, at least we'll be better prepared than most! :D


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    #14
    Challenged
    Closed Accounts Posts: 40 ✭Challenged


    It is very likely that you could be asked to find the area of the circle by integration methods as a part c question because it uses the integral which is the square root of a quadratic which needs you to make the substitution x = asinu. It has been tipped by a few teachers.


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    #15
    Cokehead Mother
    Closed Accounts Posts: 773 ✭✭✭Cokehead Mother


    Challenged wrote: »
    It is very likely that you could be asked to find the area of the circle by integration methods as a part c question because it uses the integral which is the square root of a quadratic which needs you to make the substitution x = asinu. It has been tipped by a few teachers.

    That method is kind of awkward...

    If they just want A = pi.r^2, then there's a much easier way.

    Basically you split the circle into an infinite number of circumfrences of radius x and width dx. So the area of the infinitely small circumfrences is gonna be 2pi.x.dx.

    Then the sum of all these circumferences is gonna be the integral from 0 to r of 2pi.x.dx, which is of course pi.r^2.

    It sounds awkward but it comes out in one line.

    And if the question is in the same form as the 1999 integration you can just draw it and get it out pretty quick too, without this subsitution business.


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    #16
    Parsley
    Closed Accounts Posts: 3,144 ✭✭✭Parsley


    ^^ That's the way I'd do it too, coke. Helps if you've done the applied maths proofs for moments of inertia.


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    #17
    orangetictac
    Registered Users Posts: 252 ✭✭orangetictac


    Parsley wrote: »
    It'll most likely be volume of a sphere, cone or cylinder.

    Cylinder?? help


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    #18
    Cokehead Mother
    Closed Accounts Posts: 773 ✭✭✭Cokehead Mother


    orangetictac wrote: »
    Cylinder?? help
    Cokehead Mother wrote: »
    Well you've got a line with two points, (0,r) and (h,r).

    So y = r.

    y^2 = r^2

    Integrate that from 0 to h and multiply it by pi.

    :)


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    #19
    Parsley
    Closed Accounts Posts: 3,144 ✭✭✭Parsley


    It was explained a few posts up!


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    #20
    Peleus
    Registered Users Posts: 784 ✭✭✭Peleus


    Cokehead Mother wrote: »
    That method is kind of awkward...

    If they just want A = pi.r^2, then there's a much easier way.

    Basically you split the circle into an infinite number of circumfrences of radius x and width dx. So the area of the infinitely small circumfrences is gonna be 2pi.x.dx.

    Then the sum of all these circumferences is gonna be the integral from 0 to r of 2pi.x.dx, which is of course pi.r^2.

    It sounds awkward but it comes out in one line.

    And if the question is in the same form as the 1999 integration you can just draw it and get it out pretty quick too, without this subsitution business.

    ok i got it now. Ye that adding the strips of the circumference thing works well. just like the moment of inertia proof in Ap. Maths.


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    #21
    meeka
    Registered Users Posts: 252 ✭✭meeka


    Huh, I thought that the volume of a cone and the volume of a sphere were the only ones they could ask.. But a cylinder too? Better go look this up..


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    #22
    Peleus
    Registered Users Posts: 784 ✭✭✭Peleus


    meeka wrote: »
    Huh, I thought that the volume of a cone and the volume of a sphere were the only ones they could ask.. But a cylinder too? Better go look this up..

    that one is easy. you just draw the straight line y=r in the top right quadrant. hard to explain. but easy
    then rotate around the axis. have ot look over that myself actually.

    think this is the formula: S is int sign

    Area= pi S (y^2=) dx lims are from 0 to h

    then A = pi S r^2 dx

    A= pi.r^2.h


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    #23
    Cokehead Mother
    Closed Accounts Posts: 773 ✭✭✭Cokehead Mother


    k3sy9.jpg

    The syllabus actually says cone and sphere, so they actually can't ask the cyllinder at all. My book said any line segment! :eek:

    Also while we need to integrate root(a^2 - x^2), the syllabus doesn't call them circular integrals so I guess that means they don't expect us to make that connection, and so they're not gonna ask us to prove the formula of a circle.


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    #24
    meeka
    Registered Users Posts: 252 ✭✭meeka


    ^Aha! Thought so.

    Thanks anyway Peleus, I might as well try it out anyway :)


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    #25
    cian1500ww
    Registered Users Posts: 2,904 ✭✭✭cian1500ww


    You could also be asked for a sphere or a cone.


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    #26
    Peleus
    Registered Users Posts: 784 ✭✭✭Peleus


    Cokehead Mother wrote: »
    k3sy9.jpg

    The syllabus actually says cone and sphere, so they actually can't ask the cyllinder at all. My book said any line segment! :eek:

    Also while we need to integrate root(a^2 - x^2), the syllabus doesn't call them circular integrals so I guess that means they don't expect us to make that connection, and so they're not gonna ask us to prove the formula of a circle.

    aw cool. makes life easier. only thing you really need to know is the formula for rotating around the axis, and then look up the log tables for the area your trying to prove. then just a bit of reverse engineering.

    oh yeah, off topic but i was thinking... you know when you're asked to prove something and you're like a minus sign off it and you cant see why. you could easily just put in the minus sign (or whatever you are off by) further up the sum and when you get the 'right' answer underline it loads and write QED in big letters. even tho it's wrong in the method, the examiner thinks you got the right answer so will give you full marks. will only work sometimes tho. i plan on doing that if i'm stuck.


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